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|
||||
<li class="toc-item toc-section">
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||||
<div class="toc-title-box"><a href="notes-01-13.html" class="internal"><span class="title">Tuesday, Jan 13</span></a></div>
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<ul class="structural toc-item-list">
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||||
<li class="toc-item toc-subsection">
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<div class="toc-title-box"><a href="notes-week-01.html#notes-week-01-3" class="internal"><span class="title">Tuesday 1/13</span></a></div>
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<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Sets" class="internal"><span class="title">Sec 1.1: Sets</span></a></div></li>
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Probability" class="internal"><span class="title">Sec 1.2: Probability</span></a></div></li>
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<li class="toc-item toc-subsection">
|
||||
<div class="toc-title-box"><a href="notes-week-01.html#notes-week-01-4" class="internal"><span class="title">Thursday 1/15</span></a></div>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-15.html" class="internal"><span class="title">Thursday, Jan 15</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Conditional-Probability" class="internal"><span class="title">Conditional Probability</span></a></div></li>
|
||||
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<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></div>
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||||
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<a class="previous-button button" href="syllabus.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Prev</span></a><a class="top-button button" href="#" title="Top"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Top</span></a><a class="next-button button" href="notes-01-13.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></a>
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@@ -163,26 +174,31 @@ eBookConfig.enable_chatcodes = false;
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<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="ch-notes.html" class="internal"><span class="title">Class Notes</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-section">
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<div class="toc-title-box"><a href="notes-week-01.html" class="internal"><span class="title">Week 1</span></a></div>
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<ul class="structural toc-item-list">
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<li class="toc-item toc-subsection">
|
||||
<div class="toc-title-box"><a href="notes-week-01.html#notes-week-01-3" class="internal"><span class="title">Tuesday 1/13</span></a></div>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-13.html" class="internal"><span class="title">Tuesday, Jan 13</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Sets" class="internal"><span class="title">Sec 1.1: Sets</span></a></div></li>
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Probability" class="internal"><span class="title">Sec 1.2: Probability</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-13.html#subsec-Sets" class="internal"><span class="title">Sec 1.1: Sets</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-13.html#subsec-Probability" class="internal"><span class="title">Sec 1.2: Probability</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-subsection">
|
||||
<div class="toc-title-box"><a href="notes-week-01.html#notes-week-01-4" class="internal"><span class="title">Thursday 1/15</span></a></div>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-15.html" class="internal"><span class="title">Thursday, Jan 15</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Conditional-Probability" class="internal"><span class="title">Conditional Probability</span></a></div></li>
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Diagnostic-Testing" class="internal"><span class="title">Diagnostic Testing</span></a></div></li>
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Independent-Events" class="internal"><span class="title">Independent Events</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Conditional-Probability" class="internal"><span class="title">Sec 1.3: Conditional Probability</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Diagnostic-Testing" class="internal"><span class="title">Diagnostic Testing</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Independent-Events" class="internal"><span class="title">Sec 1.4: Independent Events</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-20.html#subsec-Discrete-Random-Variables" class="internal"><span class="title">Sec 2.1: Random Variables</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="quizzes.html" class="internal"><span class="title">Quizzes</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-01.html" class="internal"><span class="title">Quiz 1</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="recitations.html" class="internal"><span class="title">Recitations</span></a></div>
|
||||
|
||||
+31
-15
@@ -101,15 +101,15 @@ eBookConfig.allow_pairs = false;
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||||
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||||
</head>
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||||
@@ -140,6 +140,17 @@ eBookConfig.enable_chatcodes = false;
|
||||
\newcommand{\Z}{\mathbb Z}
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||||
\newcommand{\Q}{\mathbb Q}
|
||||
\newcommand{\R}{\mathbb R}
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||||
|
||||
|
||||
\DeclareMathOperator{\Bin}{Bin}
|
||||
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|
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\DeclareMathOperator{\Exp}{Exp}
|
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|
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|
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\DeclareMathOperator{\E}{E}
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\DeclareMathOperator{\Var}{Var}
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\DeclareMathOperator{\Cov}{Cov}
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||||
@@ -163,26 +174,31 @@ eBookConfig.enable_chatcodes = false;
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||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="ch-notes.html" class="internal"><span class="title">Class Notes</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-week-01.html" class="internal"><span class="title">Week 1</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection">
|
||||
<div class="toc-title-box"><a href="notes-week-01.html#notes-week-01-3" class="internal"><span class="title">Tuesday 1/13</span></a></div>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-13.html" class="internal"><span class="title">Tuesday, Jan 13</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Sets" class="internal"><span class="title">Sec 1.1: Sets</span></a></div></li>
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Probability" class="internal"><span class="title">Sec 1.2: Probability</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-13.html#subsec-Sets" class="internal"><span class="title">Sec 1.1: Sets</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-13.html#subsec-Probability" class="internal"><span class="title">Sec 1.2: Probability</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-subsection">
|
||||
<div class="toc-title-box"><a href="notes-week-01.html#notes-week-01-4" class="internal"><span class="title">Thursday 1/15</span></a></div>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-15.html" class="internal"><span class="title">Thursday, Jan 15</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Conditional-Probability" class="internal"><span class="title">Conditional Probability</span></a></div></li>
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Diagnostic-Testing" class="internal"><span class="title">Diagnostic Testing</span></a></div></li>
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Independent-Events" class="internal"><span class="title">Independent Events</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Conditional-Probability" class="internal"><span class="title">Sec 1.3: Conditional Probability</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Diagnostic-Testing" class="internal"><span class="title">Diagnostic Testing</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Independent-Events" class="internal"><span class="title">Sec 1.4: Independent Events</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-20.html#subsec-Discrete-Random-Variables" class="internal"><span class="title">Sec 2.1: Random Variables</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="quizzes.html" class="internal"><span class="title">Quizzes</span></a></div>
|
||||
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|
||||
</li>
|
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<li class="toc-item toc-chapter">
|
||||
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<div class="autopermalink" data-description="Paragraph"><a href="#example-rolls-conditional-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Example 25"><a href="#example-rolls-conditional" title="Copy heading and permalink for Example 25" aria-label="Copy heading and permalink for Example 25">🔗</a></div></article><span class="incontext"><a class="internal" href="notes-week-01.html#example-rolls-conditional">in-context</a></span>
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<h1 class="heading"><a href="root-1-2.html"><span class="title">Introduction to Probability and Statistics for the Life Sciences</span> <span class="subtitle">Spring 2026 MATH 1044</span></a></h1>
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<div class="toc-title-box"><a href="notes-01-13.html" class="internal"><span class="title">Tuesday, Jan 13</span></a></div>
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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="notes-01-13"><h2 class="heading hide-type">
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<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Tuesday, Jan 13</span>
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</h2>
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<section class="introduction" id="notes-01-13-2"><div class="para" id="notes-01-13-2-1">This is an outline of the topics we covered in class. These notes are <em class="emphasis">not</em> a substitute for your own note-taking. I highly recommend that you take your own notes during class. If you ever miss a class for any reason, reach out to another student in class to get a copy of their notes.<div class="autopermalink" data-description="Paragraph"><a href="#notes-01-13-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div></section><section class="subsection" id="subsec-Sets"><h3 class="heading hide-type">
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<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Sec 1.1: Sets</span>
|
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</h3>
|
||||
<div class="para" id="subsec-Sets-2">In prior math courses, you mostly asked <dfn class="terminology">deterministic</dfn> questions. Now, we need new tools to model <dfn class="terminology">randomness</dfn>.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Sets-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<article class="example example-like" id="subsec-Sets-3"><h4 class="heading">
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||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="subsec-Sets-3-1-1">A toxin molecule in a cell has a certain chance each minute to leave the cell.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Sets-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Example 1"><a href="#subsec-Sets-3" title="Copy heading and permalink for Example 1" aria-label="Copy heading and permalink for Example 1">🔗</a></div></article><article class="example example-like" id="subsec-Sets-4"><h4 class="heading">
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||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="subsec-Sets-4-1-1">A patient takes a diagnostic test for a disease and wants to know the chance that they have the disease based on the test result.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Sets-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Example 2"><a href="#subsec-Sets-4" title="Copy heading and permalink for Example 2" aria-label="Copy heading and permalink for Example 2">🔗</a></div></article><article class="definition definition-like" id="def-sample-space"><h4 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="def-sample-space-1-1">The <dfn class="terminology">sample space</dfn>, often denoted <span class="process-math">\(\Omega\text{,}\)</span> is the set of all possible results of an experiment. A single result is called an <dfn class="terminology">outcome</dfn>, while a collection of results is called an <dfn class="terminology">event</dfn>.<div class="autopermalink" data-description="Paragraph"><a href="#def-sample-space-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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||||
<div class="autopermalink" data-description="Definition 3"><a href="#def-sample-space" title="Copy heading and permalink for Definition 3" aria-label="Copy heading and permalink for Definition 3">🔗</a></div></article><article class="example example-like" id="example-sample-space"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="example-sample-space-1-1">An experiment consists of rolling a standard 6-sided die (D6). The sample space is <span class="process-math">\(\Omega = \{1, 2, 3, 4, 5, 6\}\text{.}\)</span> One possible event is <span class="process-math">\(A = \{2, 4, 6\}\text{,}\)</span> i.e., the event that the result of the roll is even.<div class="autopermalink" data-description="Paragraph"><a href="#example-sample-space-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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||||
<div class="autopermalink" data-description="Example 4"><a href="#example-sample-space" title="Copy heading and permalink for Example 4" aria-label="Copy heading and permalink for Example 4">🔗</a></div></article><div class="para" id="subsec-Sets-7">Note: we’ll use notation like D6 to indicate a 6-sided die with faces 1, 2, 3, 4, 5, 6. Similarly, for example, D4 will indicate a 4-sided die with faces 1, 2, 3, 4.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Sets-7" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<article class="definition definition-like" id="def-subset"><h4 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">5</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="def-subset-1-1">The symbol <span class="process-math">\(\in\)</span> means "is an <dfn class="terminology">element</dfn> of", as in <span class="process-math">\(4 \in A\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-subset-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para" id="def-subset-1-2">The symbol <span class="process-math">\(\subset\)</span> means "is a <dfn class="terminology">subset</dfn> of", as in <span class="process-math">\(A \subset \Omega\text{.}\)</span> This means that every element of the set <span class="process-math">\(A\)</span> is also an element of the set <span class="process-math">\(\Omega\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-subset-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 5"><a href="#def-subset" title="Copy heading and permalink for Definition 5" aria-label="Copy heading and permalink for Definition 5">🔗</a></div></article><article class="definition definition-like" id="def-set-operations"><h4 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">6</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="def-set-operations-1-1">
|
||||
<div class="para">Consider sets <span class="process-math">\(A\)</span> and <span class="process-math">\(B\text{,}\)</span> each contained inside <span class="process-math">\(\Omega\text{.}\)</span> We can combine sets in a variety of ways:</div>
|
||||
<dl class="description-list">
|
||||
<dt id="def-set-operations-1-1-4-1">Union<div class="autopermalink" data-description="Item: Union"><a href="#def-set-operations-1-1-4-1" title="Copy heading and permalink for Item: Union" aria-label="Copy heading and permalink for Item: Union">🔗</a></div>
|
||||
</dt>
|
||||
<dd><div class="para" id="def-set-operations-1-1-4-1-2">The <dfn class="terminology">union</dfn> of <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> is the set <span class="process-math">\(A \cup B = \{x \mid x \in A \text{ or } x \in B\}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-set-operations-1-1-4-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></dd>
|
||||
<dt id="def-set-operations-1-1-4-2">Intersection<div class="autopermalink" data-description="Item: Intersection"><a href="#def-set-operations-1-1-4-2" title="Copy heading and permalink for Item: Intersection" aria-label="Copy heading and permalink for Item: Intersection">🔗</a></div>
|
||||
</dt>
|
||||
<dd><div class="para" id="def-set-operations-1-1-4-2-2">The <dfn class="terminology">intersection</dfn> of <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> is the set <span class="process-math">\(A \cap B = \{x \mid x \in A \text{ and } x \in B\}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-set-operations-1-1-4-2-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></dd>
|
||||
<dt id="def-set-operations-1-1-4-3">Difference<div class="autopermalink" data-description="Item: Difference"><a href="#def-set-operations-1-1-4-3" title="Copy heading and permalink for Item: Difference" aria-label="Copy heading and permalink for Item: Difference">🔗</a></div>
|
||||
</dt>
|
||||
<dd><div class="para" id="def-set-operations-1-1-4-3-2">The <dfn class="terminology">set difference</dfn> <span class="process-math">\(A-B\)</span> is the set <span class="process-math">\(A - B = \{x \mid x \in A \text{ and } x \notin B\}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-set-operations-1-1-4-3-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></dd>
|
||||
<dt id="def-set-operations-1-1-4-4">Complement<div class="autopermalink" data-description="Item: Complement"><a href="#def-set-operations-1-1-4-4" title="Copy heading and permalink for Item: Complement" aria-label="Copy heading and permalink for Item: Complement">🔗</a></div>
|
||||
</dt>
|
||||
<dd><div class="para" id="def-set-operations-1-1-4-4-2">The <dfn class="terminology">complement</dfn> of <span class="process-math">\(A\)</span> is the set <span class="process-math">\(A^c = \{x \in \Omega \mid x \notin A\}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-set-operations-1-1-4-4-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></dd>
|
||||
<dt id="def-set-operations-1-1-4-5">Empty Set<div class="autopermalink" data-description="Item: Empty Set"><a href="#def-set-operations-1-1-4-5" title="Copy heading and permalink for Item: Empty Set" aria-label="Copy heading and permalink for Item: Empty Set">🔗</a></div>
|
||||
</dt>
|
||||
<dd><div class="para" id="def-set-operations-1-1-4-5-2">The <dfn class="terminology">empty set</dfn>, usually written <span class="process-math">\(\emptyset\)</span> or <span class="process-math">\(\{\}\text{,}\)</span> is the set which contains no elements.<div class="autopermalink" data-description="Paragraph"><a href="#def-set-operations-1-1-4-5-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></dd>
|
||||
</dl>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#def-set-operations-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 6"><a href="#def-set-operations" title="Copy heading and permalink for Definition 6" aria-label="Copy heading and permalink for Definition 6">🔗</a></div></article><div class="para" id="subsec-Sets-10">It’s useful sometimes to draw pictures called <dfn class="terminology">Venn diagrams</dfn> representing set interactions.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Sets-10" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="sbsgroup">
|
||||
<div class="sidebyside"><div class="sbsrow" style="margin-left:5%;margin-right:5%;grid-template-columns:44.4444444444444% 44.4444444444444%;column-gap:10%;">
|
||||
<div class="sbspanel sbspanel--top top" style=""><figure class="figure figure-like" id="fig-Venn-diagram-union"><img src="generated/latex-image/fig-Venn-diagram-union-2.svg" role="img" class="contained" alt="described in detail following the image" aria-describedby="fig-Venn-diagram-union-2-description"><details class="image-description" aria-live="polite"><summary title="details"><span class="icon material-symbols-outlined" aria-hidden="true"></span></summary><div id="fig-Venn-diagram-union-2-description"><div class="para" id="fig-Venn-diagram-union-2-1-1">Venn diagram showing sets <span class="process-math">\(A, B\)</span> with the region representing <span class="process-math">\(A \cup B\)</span> shaded.<div class="autopermalink" data-description="Paragraph"><a href="#fig-Venn-diagram-union-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></div></details><figcaption><span class="type">Figure</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">7<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Union<div class="autopermalink" data-description="Figure 7"><a href="#fig-Venn-diagram-union" title="Copy heading and permalink for Figure 7" aria-label="Copy heading and permalink for Figure 7">🔗</a></div></figcaption></figure></div>
|
||||
<div class="sbspanel sbspanel--top top" style=""><figure class="figure figure-like" id="fig-Venn-diagram-intersection"><img src="generated/latex-image/fig-Venn-diagram-intersection-2.svg" role="img" class="contained" alt="described in detail following the image" aria-describedby="fig-Venn-diagram-intersection-2-description"><details class="image-description" aria-live="polite"><summary title="details"><span class="icon material-symbols-outlined" aria-hidden="true"></span></summary><div id="fig-Venn-diagram-intersection-2-description"><div class="para" id="fig-Venn-diagram-intersection-2-1-1">Venn diagram showing sets <span class="process-math">\(A, B\)</span> with the region representing <span class="process-math">\(A \cap B\)</span> shaded.<div class="autopermalink" data-description="Paragraph"><a href="#fig-Venn-diagram-intersection-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></div></details><figcaption><span class="type">Figure</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">8<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Intersection<div class="autopermalink" data-description="Figure 8"><a href="#fig-Venn-diagram-intersection" title="Copy heading and permalink for Figure 8" aria-label="Copy heading and permalink for Figure 8">🔗</a></div></figcaption></figure></div>
|
||||
</div></div>
|
||||
<div class="sidebyside"><div class="sbsrow" style="margin-left:5%;margin-right:5%;grid-template-columns:44.4444444444444% 44.4444444444444%;column-gap:10%;">
|
||||
<div class="sbspanel sbspanel--top top" style=""><figure class="figure figure-like" id="fig-Venn-diagram-difference"><img src="generated/latex-image/fig-Venn-diagram-difference-2.svg" role="img" class="contained" alt="described in detail following the image" aria-describedby="fig-Venn-diagram-difference-2-description"><details class="image-description" aria-live="polite"><summary title="details"><span class="icon material-symbols-outlined" aria-hidden="true"></span></summary><div id="fig-Venn-diagram-difference-2-description"><div class="para" id="fig-Venn-diagram-difference-2-1-1">Venn diagram showing sets <span class="process-math">\(A, B\)</span> with the region representing <span class="process-math">\(A - B\)</span> shaded.<div class="autopermalink" data-description="Paragraph"><a href="#fig-Venn-diagram-difference-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></div></details><figcaption><span class="type">Figure</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">9<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Difference<div class="autopermalink" data-description="Figure 9"><a href="#fig-Venn-diagram-difference" title="Copy heading and permalink for Figure 9" aria-label="Copy heading and permalink for Figure 9">🔗</a></div></figcaption></figure></div>
|
||||
<div class="sbspanel sbspanel--top top" style=""><figure class="figure figure-like" id="fig-Venn-diagram-complement"><img src="generated/latex-image/fig-Venn-diagram-complement-2.svg" role="img" class="contained" alt="described in detail following the image" aria-describedby="fig-Venn-diagram-complement-2-description"><details class="image-description" aria-live="polite"><summary title="details"><span class="icon material-symbols-outlined" aria-hidden="true"></span></summary><div id="fig-Venn-diagram-complement-2-description"><div class="para" id="fig-Venn-diagram-complement-2-1-1">Venn diagram showing set <span class="process-math">\(A \subset \Omega\)</span> with the region representing <span class="process-math">\(A^c\)</span> shaded.<div class="autopermalink" data-description="Paragraph"><a href="#fig-Venn-diagram-complement-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></div></details><figcaption><span class="type">Figure</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">10<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Complement<div class="autopermalink" data-description="Figure 10"><a href="#fig-Venn-diagram-complement" title="Copy heading and permalink for Figure 10" aria-label="Copy heading and permalink for Figure 10">🔗</a></div></figcaption></figure></div>
|
||||
</div></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Subsection: Sec 1.1: Sets"><a href="#subsec-Sets" title="Copy heading and permalink for Subsection: Sec 1.1: Sets" aria-label="Copy heading and permalink for Subsection: Sec 1.1: Sets">🔗</a></div></section><section class="subsection" id="subsec-Probability"><h3 class="heading hide-type">
|
||||
<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Sec 1.2: Probability</span>
|
||||
</h3>
|
||||
<div class="para" id="subsec-Probability-2">Next, we want to start assigning probabilities to each individual outcome so we can then find the probabilities of events.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Probability-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="example example-like" id="subsec-Probability-3"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">11</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="subsec-Probability-3-1-1">An experiment consists of rolling a D6. The sample space is <span class="process-math">\(\Omega = \{1, 2, 3, 4, 5, 6\}\text{.}\)</span> We might assign probabilities as follows:<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Probability-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <figure class="table table-like" id="subsec-Probability-3-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">12<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Distribution for a fair die<div class="autopermalink" data-description="Table 12: Distribution for a fair die"><a href="#subsec-Probability-3-1-2" title="Copy heading and permalink for Table 12: Distribution for a fair die" aria-label="Copy heading and permalink for Table 12: Distribution for a fair die">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">1</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">2</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">3</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">4</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">5</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">6</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
</table></div></figure> <div class="para logical" id="subsec-Probability-3-1-3">
|
||||
<div class="para">Note that we don’t have to assign the same probability to each outcome. If we do, we call this distribution <dfn class="terminology">uniform</dfn>. If we have some event, such as <span class="process-math">\(A = \{2, 4, 6\}\text{,}\)</span> then we calculate the probability of the event by adding together the probabilities of the outcomes that make up the event:</div>
|
||||
<div class="displaymath process-math" id="subsec-Probability-3-1-3-3">
|
||||
\begin{gather*}
|
||||
\Pr(A) = \Pr(2) + \Pr(4) + \Pr(6) = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">Note that this is the same result that we would get if we counted the total number of outcomes in <span class="process-math">\(A\)</span> and divided by the total number of outcomes in <span class="process-math">\(\Omega\text{.}\)</span>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Probability-3-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 11"><a href="#subsec-Probability-3" title="Copy heading and permalink for Example 11" aria-label="Copy heading and permalink for Example 11">🔗</a></div></article><article class="definition definition-like" id="def-cardinality"><h4 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">13</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="def-cardinality-1-1">Given a finite set <span class="process-math">\(A\text{,}\)</span> the <dfn class="terminology">cardinality</dfn> of <span class="process-math">\(A\text{,}\)</span> written <span class="process-math">\(|A|\text{,}\)</span> is the number of elements in <span class="process-math">\(A\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-cardinality-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 13"><a href="#def-cardinality" title="Copy heading and permalink for Definition 13" aria-label="Copy heading and permalink for Definition 13">🔗</a></div></article><article class="fact theorem-like" id="subsec-Probability-5"><h4 class="heading">
|
||||
<span class="type">Fact</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">14</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="subsec-Probability-5-1-1">
|
||||
<div class="para">If <span class="process-math">\(\Omega\)</span> is a finite probability space with the uniform distribution and <span class="process-math">\(A \subset \Omega\)</span> is an event, then:</div>
|
||||
<div class="displaymath process-math" id="subsec-Probability-5-1-1-3">
|
||||
\begin{gather*}
|
||||
\Pr(A) = \frac{|A|}{|\Omega|}
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Probability-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Fact 14"><a href="#subsec-Probability-5" title="Copy heading and permalink for Fact 14" aria-label="Copy heading and permalink for Fact 14">🔗</a></div></article><article class="example example-like" id="subsec-Probability-6"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">15</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="subsec-Probability-6-1-1">Suppose we have a weighted die that’s much more likely to come up 6 than any other outcome.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Probability-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <figure class="table table-like" id="subsec-Probability-6-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">16<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Distribution for a fair die<div class="autopermalink" data-description="Table 16: Distribution for a fair die"><a href="#subsec-Probability-6-1-2" title="Copy heading and permalink for Table 16: Distribution for a fair die" aria-label="Copy heading and permalink for Table 16: Distribution for a fair die">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">1</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.1</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">2</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.1</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">3</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.1</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">4</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.2</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">5</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.1</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">6</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.4</td>
|
||||
</tr>
|
||||
</table></div></figure> <div class="para logical" id="subsec-Probability-6-1-3">
|
||||
<div class="para">With this distribution, if <span class="process-math">\(A = \{2, 4, 6\}\text{,}\)</span> then:</div>
|
||||
<div class="displaymath process-math" id="subsec-Probability-6-1-3-2">
|
||||
\begin{gather*}
|
||||
\Pr(A) = 0.1 + 0.2 + 0.4 = 0.7 \neq \frac{3}{6}.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Probability-6-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 15"><a href="#subsec-Probability-6" title="Copy heading and permalink for Example 15" aria-label="Copy heading and permalink for Example 15">🔗</a></div></article><article class="example example-like" id="subsec-Probability-7"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">17</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="subsec-Probability-7-1-1">Pick a number from 1 to 10. What is the probability of picking 3?<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Probability-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para logical" id="subsec-Probability-7-1-2">
|
||||
<div class="para">Some considerations:</div>
|
||||
<ol class="decimal" id="subsec-Probability-7-1-2-1">
|
||||
<li id="subsec-Probability-7-1-2-1-1">
|
||||
<div class="para" id="subsec-Probability-7-1-2-1-1-1">Are we using the uniform distribution?<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Probability-7-1-2-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Item 1"><a href="#subsec-Probability-7-1-2-1-1" title="Copy heading and permalink for Item 1" aria-label="Copy heading and permalink for Item 1">🔗</a></div>
|
||||
</li>
|
||||
<li id="subsec-Probability-7-1-2-1-2">
|
||||
<div class="para" id="subsec-Probability-7-1-2-1-2-1">What even is the sample space here? Do we need to pick only integers, or is <span class="process-math">\(\pi\)</span> an outcome here?<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Probability-7-1-2-1-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Item 2"><a href="#subsec-Probability-7-1-2-1-2" title="Copy heading and permalink for Item 2" aria-label="Copy heading and permalink for Item 2">🔗</a></div>
|
||||
</li>
|
||||
<li id="subsec-Probability-7-1-2-1-3">
|
||||
<div class="para" id="subsec-Probability-7-1-2-1-3-1">What would "uniform" mean if the sample space was infinite?<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Probability-7-1-2-1-3-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
|
||||
<div class="autopermalink" data-description="Item 3"><a href="#subsec-Probability-7-1-2-1-3" title="Copy heading and permalink for Item 3" aria-label="Copy heading and permalink for Item 3">🔗</a></div>
|
||||
</li>
|
||||
</ol>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Probability-7-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 17"><a href="#subsec-Probability-7" title="Copy heading and permalink for Example 17" aria-label="Copy heading and permalink for Example 17">🔗</a></div></article><article class="definition definition-like" id="def-probability-distribution"><h4 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">18</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="def-probability-distribution-1-1">
|
||||
<div class="para">A <dfn class="terminology">probability distribution</dfn> on a sample space <span class="process-math">\(\Omega\)</span> assigns probabilities to every event, satisfying the following conditions:</div>
|
||||
<ol class="decimal" id="def-probability-distribution-1-1-3">
|
||||
<li id="def-probability-distribution-1-1-3-1">
|
||||
<div class="para" id="def-probability-distribution-1-1-3-1-1">
|
||||
<span class="process-math">\(\Pr(\Omega) = 1\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-probability-distribution-1-1-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Item 1"><a href="#def-probability-distribution-1-1-3-1" title="Copy heading and permalink for Item 1" aria-label="Copy heading and permalink for Item 1">🔗</a></div>
|
||||
</li>
|
||||
<li id="def-probability-distribution-1-1-3-2">
|
||||
<div class="para" id="def-probability-distribution-1-1-3-2-1">
|
||||
<span class="process-math">\(0 \leq \Pr(A) \leq 1\)</span> for any event <span class="process-math">\(A\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-probability-distribution-1-1-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Item 2"><a href="#def-probability-distribution-1-1-3-2" title="Copy heading and permalink for Item 2" aria-label="Copy heading and permalink for Item 2">🔗</a></div>
|
||||
</li>
|
||||
<li id="def-probability-distribution-1-1-3-3">
|
||||
<div class="para" id="def-probability-distribution-1-1-3-3-1">If <span class="process-math">\(A \cap B = \emptyset\text{,}\)</span> then <span class="process-math">\(\Pr(A\cup B) = \Pr(A) + \Pr(B)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-probability-distribution-1-1-3-3-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Item 3"><a href="#def-probability-distribution-1-1-3-3" title="Copy heading and permalink for Item 3" aria-label="Copy heading and permalink for Item 3">🔗</a></div>
|
||||
</li>
|
||||
</ol>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#def-probability-distribution-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 18"><a href="#def-probability-distribution" title="Copy heading and permalink for Definition 18" aria-label="Copy heading and permalink for Definition 18">🔗</a></div></article><article class="example example-like" id="subsec-Probability-9"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">19</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="subsec-Probability-9-1-1">Suppose we flip a coin until we see heads. The sample space is <span class="process-math">\(\Omega = \{H, TH, TTH, TTTH, \dotsc\}\text{.}\)</span> What would "uniform" mean here? Is there any way we could assign the same probability to each individual outcome here?<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Probability-9-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para" id="subsec-Probability-9-1-2">Instead of "uniform", what if we want to treat the coin as a fair coin? Then what would the distribution be?<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Probability-9-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <figure class="table table-like" id="subsec-Probability-9-1-3"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">20<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Distribution assuming a fair coin<div class="autopermalink" data-description="Table 20: Distribution assuming a fair coin"><a href="#subsec-Probability-9-1-3" title="Copy heading and permalink for Table 20: Distribution assuming a fair coin" aria-label="Copy heading and permalink for Table 20: Distribution assuming a fair coin">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"># flips</td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(H\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines">1</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/2\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(TH\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines">2</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/4\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(TTH\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines">3</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/8\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(TTTH\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines">4</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/16\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(T\dotsm TH\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(n\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/2^n\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
|
||||
</tr>
|
||||
</table></div></figure><div class="autopermalink" data-description="Example 19"><a href="#subsec-Probability-9" title="Copy heading and permalink for Example 19" aria-label="Copy heading and permalink for Example 19">🔗</a></div></article><article class="example example-like" id="subsec-Probability-10"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">21</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="subsec-Probability-10-1-1">One last (very useful!) observation. Recall: If <span class="process-math">\(A\cap B = \emptyset\text{,}\)</span> then <span class="process-math">\(\Pr(A\cup B) = \Pr(A) + \Pr(B)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#subsec-Probability-10-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para logical" id="subsec-Probability-10-1-2">
|
||||
<div class="para">For any event <span class="process-math">\(A\text{,}\)</span> <span class="process-math">\(A\cap A^c = \emptyset\)</span> and <span class="process-math">\(A \cup A^c = \Omega\text{.}\)</span> So:</div>
|
||||
<div class="displaymath process-math" id="subsec-Probability-10-1-2-4">
|
||||
\begin{gather*}
|
||||
1 = \Pr(\underbrace{A \cup A^c}_{\Omega}) = \Pr(A) + \Pr(A^c).
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">We can rewrite this in two useful ways:</div>
|
||||
<div class="displaymath process-math" id="subsec-Probability-10-1-2-5">
|
||||
\begin{align*}
|
||||
\Pr(A) \amp = 1 - \Pr(A^c) \\
|
||||
\Pr(A^c) \amp = 1 - \Pr(A)
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Probability-10-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
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||||
<div class="autopermalink" data-description="Example 21"><a href="#subsec-Probability-10" title="Copy heading and permalink for Example 21" aria-label="Copy heading and permalink for Example 21">🔗</a></div></article><div class="autopermalink" data-description="Subsection: Sec 1.2: Probability"><a href="#subsec-Probability" title="Copy heading and permalink for Subsection: Sec 1.2: Probability" aria-label="Copy heading and permalink for Subsection: Sec 1.2: Probability">🔗</a></div></section><div class="autopermalink" data-description="Section: Tuesday, Jan 13"><a href="#notes-01-13" title="Copy heading and permalink for Section: Tuesday, Jan 13" aria-label="Copy heading and permalink for Section: Tuesday, Jan 13">🔗</a></div></section></div>
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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="notes-01-15"><h2 class="heading hide-type">
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<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Thursday, Jan 15</span>
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</h2>
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<section class="introduction" id="notes-01-15-2"><div class="para" id="notes-01-15-2-1">This is an outline of the topics we covered in class. These notes are <em class="emphasis">not</em> a substitute for your own note-taking. I highly recommend that you take your own notes during class. If you ever miss a class for any reason, reach out to another student in class to get a copy of their notes.<div class="autopermalink" data-description="Paragraph"><a href="#notes-01-15-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div></section><section class="subsection" id="subsec-Conditional-Probability"><h3 class="heading hide-type">
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<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Sec 1.3: Conditional Probability</span>
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||||
</h3>
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||||
<div class="para" id="subsec-Conditional-Probability-2">Question: How does evidence (e.g., knowledge of one event occurring) change our knowledge of probabilities for other events?<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Conditional-Probability-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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||||
<article class="example example-like" id="subsec-Conditional-Probability-3"><h4 class="heading">
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||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">22</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
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||||
<div class="para" id="subsec-Conditional-Probability-3-1-1">Roll a fair D6 two times. Let <span class="process-math">\(A = \{\text{sum } \geq 10\}\)</span> and <span class="process-math">\(B = \{\text{first roll is } 6\}\text{.}\)</span> <span class="process-math">\(A\)</span> feels more likely if we already know <span class="process-math">\(B\)</span> has occurred.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Conditional-Probability-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Example 22"><a href="#subsec-Conditional-Probability-3" title="Copy heading and permalink for Example 22" aria-label="Copy heading and permalink for Example 22">🔗</a></div></article><article class="definition definition-like" id="def-conditional-probability"><h4 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">23</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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||||
</h4>
|
||||
<div class="para logical" id="def-conditional-probability-1-1">
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||||
<div class="para">The <dfn class="terminology">conditional probability</dfn> of <span class="process-math">\(A\)</span> given <span class="process-math">\(B\)</span> is: ,</div>
|
||||
<div class="displaymath process-math" id="def-conditional-probability-1-1-4">
|
||||
\begin{gather*}
|
||||
\Pr(A \mid B) = \frac{\Pr(A\cap B)}{\Pr(B)}
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#def-conditional-probability-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <figure class="figure figure-like" id="fig-conditional-probability"><div class="image-box" style="width: 50%; margin-left: 25%; margin-right: 25%;">
|
||||
<img src="generated/latex-image/fig-conditional-probability-2.svg" role="img" class="contained" alt="described in detail following the image" aria-describedby="fig-conditional-probability-2-description"><details class="image-description" aria-live="polite"><summary title="details"><span class="icon material-symbols-outlined" aria-hidden="true"></span></summary><div id="fig-conditional-probability-2-description"><div class="para" id="fig-conditional-probability-2-1-1">Two overlapping circles representing events <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> sit inside a rectangle representing the sample space <span class="process-math">\(\Omega\text{.}\)</span> The circle labeled <span class="process-math">\(B\)</span> is shaded. The portion of that circle which is overlapped by the <span class="process-math">\(A\)</span> circle is also filled in with slanted lines.<div class="autopermalink" data-description="Paragraph"><a href="#fig-conditional-probability-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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||||
</div></div></details>
|
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||||
<figcaption><span class="type">Figure</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">24<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="process-math">\(\Pr(A \mid B)\)</span> tells the proportion of <span class="process-math">\(B\)</span> which is overlapped by <span class="process-math">\(A\text{.}\)</span><div class="autopermalink" data-description="Figure 24"><a href="#fig-conditional-probability" title="Copy heading and permalink for Figure 24" aria-label="Copy heading and permalink for Figure 24">🔗</a></div></figcaption></figure><div class="autopermalink" data-description="Definition 23"><a href="#def-conditional-probability" title="Copy heading and permalink for Definition 23" aria-label="Copy heading and permalink for Definition 23">🔗</a></div></article><article class="example example-like" id="example-rolls-conditional"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">25</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="example-rolls-conditional-1-1">
|
||||
<div class="para">Continuing from the previous example, <span class="process-math">\(|\Omega| = 36\text{.}\)</span>
|
||||
</div>
|
||||
<div class="displaymath process-math" id="example-rolls-conditional-1-1-2">
|
||||
\begin{align*}
|
||||
A \amp = \{(4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)\} \\
|
||||
B \amp = \{(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\} \\
|
||||
A \cap B \amp = \{(6, 4), (6, 5), (6, 6)\}
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">So <span class="process-math">\(\Pr(A) = \frac{6}{36}, \Pr(B) = \frac{6}{36}, \text{and } \Pr(A\cap B) = \frac{3}{36}\text{.}\)</span> Then:</div>
|
||||
<div class="displaymath process-math" id="example-rolls-conditional-1-1-4">
|
||||
\begin{gather*}
|
||||
\Pr(A \mid B) = \frac{\Pr(A\cap B)}{\Pr(B)} = \frac{3/36}{6/36} = \frac{3}{6} = \frac{1}{2}.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">Notice that <span class="process-math">\(\Pr(A\mid B)\)</span> is significantly larger than <span class="process-math">\(\Pr(A)\text{.}\)</span>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#example-rolls-conditional-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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||||
</div>
|
||||
<div class="autopermalink" data-description="Example 25"><a href="#example-rolls-conditional" title="Copy heading and permalink for Example 25" aria-label="Copy heading and permalink for Example 25">🔗</a></div></article><div class="autopermalink" data-description="Subsection: Sec 1.3: Conditional Probability"><a href="#subsec-Conditional-Probability" title="Copy heading and permalink for Subsection: Sec 1.3: Conditional Probability" aria-label="Copy heading and permalink for Subsection: Sec 1.3: Conditional Probability">🔗</a></div></section><section class="subsection" id="subsec-Diagnostic-Testing"><h3 class="heading hide-type">
|
||||
<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Diagnostic Testing</span>
|
||||
</h3>
|
||||
<div class="para" id="subsec-Diagnostic-Testing-2">Setup: A patient takes a diagnostic test. Let <span class="process-math">\(P\)</span> be the event that they test positive. Let <span class="process-math">\(D\)</span> be the event that they have the disease.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Diagnostic-Testing-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="definition definition-like" id="def-sensitivity-specificity"><h4 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">26</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="def-sensitivity-specificity-1-1">The <dfn class="terminology">sensitivity</dfn> of a diagnostic test is <span class="process-math">\(\Pr(P \mid D)\text{.}\)</span> The <dfn class="terminology">specificity</dfn> of a diagnostic test is <span class="process-math">\(\Pr(P^c \mid D^c)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-sensitivity-specificity-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 26"><a href="#def-sensitivity-specificity" title="Copy heading and permalink for Definition 26" aria-label="Copy heading and permalink for Definition 26">🔗</a></div></article><div class="para" id="subsec-Diagnostic-Testing-4">But, what the patient really wants to know is <span class="process-math">\(\Pr(D \mid P)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#subsec-Diagnostic-Testing-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="example example-like" id="subsec-Diagnostic-Testing-5"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">27</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="subsec-Diagnostic-Testing-5-1-1">A disease has a prevalence of 1%. A test has sensitivity of 90% and specificity of 91%. For a patient who gets a positive test result, what is the probability that they have the disease?<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Diagnostic-Testing-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para logical" id="subsec-Diagnostic-Testing-5-1-2">
|
||||
<div class="displaymath process-math" id="subsec-Diagnostic-Testing-5-1-2-1">
|
||||
\begin{align*}
|
||||
\text{A) } 9/10 \amp \amp \text{B) } 8/10 \amp \amp \text{C) } 1/10 \amp \amp \text{D) } 1/100
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Diagnostic-Testing-5-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="subsec-Diagnostic-Testing-5-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
|
||||
<div class="para" id="subsec-Diagnostic-Testing-5-2-1">C!<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Diagnostic-Testing-5-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Answer 27.1"><a href="#subsec-Diagnostic-Testing-5-2" title="Copy heading and permalink for Answer 27.1" aria-label="Copy heading and permalink for Answer 27.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Example 27"><a href="#subsec-Diagnostic-Testing-5" title="Copy heading and permalink for Example 27" aria-label="Copy heading and permalink for Example 27">🔗</a></div></article><article class="theorem theorem-like" id="thm-Bayes-v1"><h4 class="heading">
|
||||
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">28</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Bayes’ Theorem (v1).</span>
|
||||
</h4>
|
||||
<div class="para logical" id="thm-Bayes-v1-2-1">
|
||||
<div class="para">For events <span class="process-math">\(A, B\)</span> with nonzero probability:</div>
|
||||
<div class="displaymath process-math" id="thm-Bayes-v1-2-1-2">
|
||||
\begin{gather*}
|
||||
\Pr(B \mid A) = \frac{\Pr(A \mid B)\Pr(B)}{\Pr(A)}
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#thm-Bayes-v1-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Theorem 28: Bayes’ Theorem (v1)"><a href="#thm-Bayes-v1" title="Copy heading and permalink for Theorem 28: Bayes’ Theorem (v1)" aria-label="Copy heading and permalink for Theorem 28: Bayes’ Theorem (v1)">🔗</a></div></article><div class="para logical" id="subsec-Diagnostic-Testing-7">
|
||||
<div class="para">For example, if a patient sees a positive diagnostic test result, they might try to calculate:</div>
|
||||
<div class="displaymath process-math" id="subsec-Diagnostic-Testing-7-1">
|
||||
\begin{gather*}
|
||||
\Pr(D \mid P) = \frac{\Pr(P \mid D)\Pr(D)}{\Pr(P)}
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">
|
||||
<span class="process-math">\(\Pr(P \mid D)\)</span> is the sensitivity. <span class="process-math">\(\Pr(D)\)</span> could be the prevalence. We don’t have direct access to <span class="process-math">\(\Pr(P)\text{.}\)</span>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Diagnostic-Testing-7" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="para" id="subsec-Diagnostic-Testing-8">Observation: <span class="process-math">\(\Omega = D \cup D^c\text{,}\)</span> so <span class="process-math">\(P = (P\cap D) \cup (P\cap D^c)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#subsec-Diagnostic-Testing-8" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<figure class="figure figure-like" id="fig-P-breakdown"><div class="image-box" style="width: 50%; margin-left: 25%; margin-right: 25%;">
|
||||
<img src="generated/latex-image/fig-P-breakdown-2.svg" role="img" class="contained" alt="described in detail following the image" aria-describedby="fig-P-breakdown-2-description"><details class="image-description" aria-live="polite"><summary title="details"><span class="icon material-symbols-outlined" aria-hidden="true"></span></summary><div id="fig-P-breakdown-2-description"><div class="para" id="fig-P-breakdown-2-1-1"><div class="autopermalink" data-description="Paragraph"><a href="#fig-P-breakdown-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div></div></div></details>
|
||||
</div>
|
||||
<figcaption><span class="type">Figure</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">29<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" data-description="Figure 29"><a href="#fig-P-breakdown" title="Copy heading and permalink for Figure 29" aria-label="Copy heading and permalink for Figure 29">🔗</a></div></figcaption></figure><div class="para logical" id="subsec-Diagnostic-Testing-10">
|
||||
<div class="para">Observation 2:</div>
|
||||
<div class="displaymath process-math" id="subsec-Diagnostic-Testing-10-1">
|
||||
\begin{align*}
|
||||
\Pr(P \mid D) \amp \frac{\Pr(P\cap D)}{\Pr(D)} \amp \amp \Rightarrow \amp \Pr(P \cap D) \amp = \Pr(P\mid D)\Pr(D) \\
|
||||
\Pr(P \mid D^c) \amp \frac{\Pr(P\cap D^c)}{\Pr(D^c)} \amp \amp \Rightarrow \amp \Pr(P \cap D^c) \amp = \Pr(P\mid D^c)\Pr(D^c)
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">So:</div>
|
||||
<div class="displaymath process-math" id="subsec-Diagnostic-Testing-10-2">
|
||||
\begin{gather*}
|
||||
\Pr(P) = \Pr(P\mid D)\Pr(D) + \Pr(P\mid D^c)\Pr(D^c)
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Diagnostic-Testing-10" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="theorem theorem-like" id="thm-Bayes-v2"><h4 class="heading">
|
||||
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">30</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Bayes’ Theorem (v2).</span>
|
||||
</h4>
|
||||
<div class="para logical" id="thm-Bayes-v2-2-1">
|
||||
<div class="displaymath process-math" id="thm-Bayes-v2-2-1-1">
|
||||
\begin{gather*}
|
||||
\Pr(B \mid A) = \frac{\Pr(A \mid B)\Pr(B)}{\Pr(A \mid B)\Pr(B) + \Pr(A \mid B^c)\Pr(B^c)}
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#thm-Bayes-v2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Theorem 30: Bayes’ Theorem (v2)"><a href="#thm-Bayes-v2" title="Copy heading and permalink for Theorem 30: Bayes’ Theorem (v2)" aria-label="Copy heading and permalink for Theorem 30: Bayes’ Theorem (v2)">🔗</a></div></article><article class="example example-like" id="subsec-Diagnostic-Testing-12"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">31</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="subsec-Diagnostic-Testing-12-1-1">
|
||||
<div class="para">Continuing from the previous example:</div>
|
||||
<div class="displaymath process-math" id="subsec-Diagnostic-Testing-12-1-1-1">
|
||||
\begin{align*}
|
||||
\Pr(D\mid P) \amp = \frac{\Pr(P\mid D)\Pr(D)}{\Pr(P\mid D)\Pr(D) + \Pr(P\mid D^c)\Pr(D^c)} \\
|
||||
\amp = \frac{(0.9)(0.01)}{(0.9)(0.01) + (1 - 0.91)(1 - 0.01)} \\
|
||||
\amp \approx 0.092
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">What if the patient got a negative test result instead? In that case, what is the probaiblity they do not have the disease?</div>
|
||||
<div class="displaymath process-math" id="subsec-Diagnostic-Testing-12-1-1-2">
|
||||
\begin{align*}
|
||||
\Pr(D^c\mid P^c) \amp = \frac{\Pr(P^c\mid D^c)\Pr(D^c)}{\Pr(P^c\mid D^c)\Pr(D^c) + \Pr(P^c\mid D)\Pr(D)} \\
|
||||
\amp = \frac{(0.91)(1 - 0.01)}{(0.91)(1 - 0.01) + (1 - 0.9)(0.01)} \\
|
||||
\amp \approx 0.999
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Diagnostic-Testing-12-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 31"><a href="#subsec-Diagnostic-Testing-12" title="Copy heading and permalink for Example 31" aria-label="Copy heading and permalink for Example 31">🔗</a></div></article><div class="autopermalink" data-description="Subsection: Diagnostic Testing"><a href="#subsec-Diagnostic-Testing" title="Copy heading and permalink for Subsection: Diagnostic Testing" aria-label="Copy heading and permalink for Subsection: Diagnostic Testing">🔗</a></div></section><section class="subsection" id="subsec-Independent-Events"><h3 class="heading hide-type">
|
||||
<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Sec 1.4: Independent Events</span>
|
||||
</h3>
|
||||
<div class="para" id="subsec-Independent-Events-2">Question: <span class="process-math">\(\Pr(A \mid B)\)</span> is supposed to capture how information about <span class="process-math">\(B\)</span> affects the probability of <span class="process-math">\(A\text{.}\)</span> What if it doesn’t?<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Independent-Events-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="definition definition-like" id="def-independent-events"><h4 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">32</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="def-independent-events-1-1">Events <span class="process-math">\(A, B\)</span> are <dfn class="terminology">independent</dfn> if <span class="process-math">\(\Pr(A \mid B) = \Pr(A)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-independent-events-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 32"><a href="#def-independent-events" title="Copy heading and permalink for Definition 32" aria-label="Copy heading and permalink for Definition 32">🔗</a></div></article><div class="para logical" id="subsec-Independent-Events-4">
|
||||
<div class="para">Observation: If <span class="process-math">\(A, B\)</span> have nonzero probability and are independent, then:</div>
|
||||
<div class="displaymath process-math" id="subsec-Independent-Events-4-2">
|
||||
\begin{align*}
|
||||
\Pr(A\mid B) \amp = \Pr(A) \\
|
||||
\frac{\Pr(A\cap B)}{\Pr(B)} \amp = \Pr(A) \\
|
||||
\Pr(A\cap B) \amp = \Pr(A)\Pr(B)
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">We can take this last equation as a definition of independence.</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Independent-Events-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="example example-like" id="example-rolls-independent"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">33</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="example-rolls-independent-1-1">Continuing <a href="notes-01-15.html#example-rolls-conditional" class="xref" data-knowl="./knowl/xref/example-rolls-conditional.html" data-reveal-label="Reveal" data-close-label="Close" title="Example 25">Example 25</a>, recall <span class="process-math">\(A = \{\text{sum} \geq 10\}\)</span> and <span class="process-math">\(B = \{\text{1st roll is } 6\}\text{.}\)</span> We found that <span class="process-math">\(\Pr(A \mid B) \neq \Pr(A)\text{,}\)</span> so <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> are not independent.<div class="autopermalink" data-description="Paragraph"><a href="#example-rolls-independent-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para logical" id="example-rolls-independent-1-2">
|
||||
<div class="para">Now consider the event <span class="process-math">\(C = \{\text{sum} = 7\}\text{.}\)</span> We have:</div>
|
||||
<div class="displaymath process-math" id="example-rolls-independent-1-2-2">
|
||||
\begin{align*}
|
||||
C \amp = \{(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)\} \\
|
||||
\Pr(C) \amp = \frac{6}{36} = \frac{1}{6} \\
|
||||
B\cap C \amp = \{(6, 1)\} \\
|
||||
\Pr(B\cap C) \amp = \frac{1}{36} \\
|
||||
\text{therefore: } \Pr(C\mid B) \amp = \frac{\Pr(C\cap B)}{\Pr(B)} = \frac{1/36}{1/6} = \frac{1}{6} = \Pr(C)
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">Therefore events <span class="process-math">\(B, C\)</span> are independent.</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#example-rolls-independent-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 33"><a href="#example-rolls-independent" title="Copy heading and permalink for Example 33" aria-label="Copy heading and permalink for Example 33">🔗</a></div></article><div class="autopermalink" data-description="Subsection: Sec 1.4: Independent Events"><a href="#subsec-Independent-Events" title="Copy heading and permalink for Subsection: Sec 1.4: Independent Events" aria-label="Copy heading and permalink for Subsection: Sec 1.4: Independent Events">🔗</a></div></section><div class="autopermalink" data-description="Section: Thursday, Jan 15"><a href="#notes-01-15" title="Copy heading and permalink for Section: Thursday, Jan 15" aria-label="Copy heading and permalink for Section: Thursday, Jan 15">🔗</a></div></section></div>
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<h1 class="heading"><a href="root-1-2.html"><span class="title">Introduction to Probability and Statistics for the Life Sciences</span> <span class="subtitle">Spring 2026 MATH 1044</span></a></h1>
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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="notes-01-20"><h2 class="heading hide-type">
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<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Tuesday, Jan 20</span>
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</h2>
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<section class="introduction" id="notes-01-20-2"><div class="para" id="notes-01-20-2-1">This is an outline of the topics we covered in class. These notes are <em class="emphasis">not</em> a substitute for your own note-taking. I highly recommend that you take your own notes during class. If you ever miss a class for any reason, reach out to another student in class to get a copy of their notes.<div class="autopermalink" data-description="Paragraph"><a href="#notes-01-20-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div></section><section class="subsection" id="subsec-Discrete-Random-Variables"><h3 class="heading hide-type">
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<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Sec 2.1: Random Variables</span>
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<article class="definition definition-like" id="def-RV"><h4 class="heading">
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||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">34</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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||||
</h4>
|
||||
<div class="para" id="def-RV-1-1">A <dfn class="terminology">random variable</dfn> is a function <span class="process-math">\(X \colon \Omega \to \R\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-RV-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Definition 34"><a href="#def-RV" title="Copy heading and permalink for Definition 34" aria-label="Copy heading and permalink for Definition 34">🔗</a></div></article><div class="para" id="subsec-Discrete-Random-Variables-3">The idea is that <span class="process-math">\(X\)</span> is a variable representing a real number value which depends on the outcome of an experiment.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<article class="example example-like" id="subsec-Discrete-Random-Variables-4"><h4 class="heading">
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<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">35</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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</h4>
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||||
<div class="para logical" id="subsec-Discrete-Random-Variables-4-1-1">
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||||
<div class="para">An experiment consists of planting 50 seeds in a garden, then growing them for 3 months. Let <span class="process-math">\(H_i\)</span> be the height of plant <span class="process-math">\(i\text{.}\)</span> Let <span class="process-math">\(D\)</span> be the number of seeds that didn’t sprout. Let</div>
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||||
<div class="displaymath process-math" id="subsec-Discrete-Random-Variables-4-1-1-4">
|
||||
\begin{align*}
|
||||
A \amp = \text{avg height of all 50 plants} \\
|
||||
\amp = \frac{H_1 + H_2 + \dotsb + H_{50}}{50}
|
||||
\end{align*}
|
||||
</div>
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<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Example 35"><a href="#subsec-Discrete-Random-Variables-4" title="Copy heading and permalink for Example 35" aria-label="Copy heading and permalink for Example 35">🔗</a></div></article><div class="para" id="subsec-Discrete-Random-Variables-5">A random variable has its own probability distribution.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-5" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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||||
</div>
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||||
<article class="example example-like" id="subsec-Discrete-Random-Variables-6"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">36</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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||||
</h4>
|
||||
<div class="para" id="subsec-Discrete-Random-Variables-6-1-1">Roll a fair D6 twice. Let <span class="process-math">\(S\)</span> be the sum of the rolls. Then <span class="process-math">\(\Omega = \{(1, 1), (1, 2), \dotsc, (6, 6)\}\)</span> has 36 elements. Since the die is fair, the distribution on <span class="process-math">\(\Omega\)</span> is uniform, i.e., <span class="process-math">\(\Pr(\omega) = \frac{1}{36}\)</span> for any <span class="process-math">\(\omega \in \Omega\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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||||
</div> <div class="para" id="subsec-Discrete-Random-Variables-6-1-2">
|
||||
<span class="process-math">\(S\)</span> takes on the values <span class="process-math">\(2, 3, 4, \dotsc, 12\text{,}\)</span> with probabilities:<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-6-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <figure class="table table-like" id="subsec-Discrete-Random-Variables-6-1-3"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">37<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Distribution for <span class="process-math">\(S\)</span><div class="autopermalink" data-description="Table 37: Distribution for S"><a href="#subsec-Discrete-Random-Variables-6-1-3" title="Copy heading and permalink for Table 37: Distribution for S" aria-label="Copy heading and permalink for Table 37: Distribution for S">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(x\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\Pr(S = x)\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">2</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">1/36</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">3</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">2/36</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">4</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">3/36</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">7</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">6/36</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">8</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">5/36</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">12</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">1/36</td>
|
||||
</tr>
|
||||
</table></div></figure> <div class="para" id="subsec-Discrete-Random-Variables-6-1-4">Note that the distribution on <span class="process-math">\(\Omega\)</span> is uniform, but the distribution on <span class="process-math">\(S\)</span> is not.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-6-1-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 36"><a href="#subsec-Discrete-Random-Variables-6" title="Copy heading and permalink for Example 36" aria-label="Copy heading and permalink for Example 36">🔗</a></div></article><article class="example example-like" id="subsec-Discrete-Random-Variables-7"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">38</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="subsec-Discrete-Random-Variables-7-1-1">
|
||||
<div class="para">Let <span class="process-math">\(\Omega\)</span> be a sample space and <span class="process-math">\(A \subset \Omega\)</span> an event. Let</div>
|
||||
<div class="displaymath process-math" id="subsec-Discrete-Random-Variables-7-1-1-3">
|
||||
\begin{align*}
|
||||
X = \begin{cases} 1 \amp x \in A \\ 0 \amp x \notin A \end{cases}
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">
|
||||
<span class="process-math">\(X\)</span> is called an <dfn class="terminology">indicator random variable</dfn>, and we say "<span class="process-math">\(X\)</span> <dfn class="terminology">indicates</dfn> <span class="process-math">\(A\)</span>".</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para logical" id="subsec-Discrete-Random-Variables-7-1-2">
|
||||
<div class="para">The distribution on <span class="process-math">\(X\)</span> is:</div>
|
||||
<div class="displaymath process-math" id="subsec-Discrete-Random-Variables-7-1-2-2">
|
||||
\begin{align*}
|
||||
\Pr(X = 1) \amp = \Pr(\{ x \mid x \in A\}) = \Pr(A) \\
|
||||
\Pr(X = 0) \amp = 1 - \Pr(A)
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-7-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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||||
</div>
|
||||
<div class="autopermalink" data-description="Example 38"><a href="#subsec-Discrete-Random-Variables-7" title="Copy heading and permalink for Example 38" aria-label="Copy heading and permalink for Example 38">🔗</a></div></article><article class="example example-like" id="subsec-Discrete-Random-Variables-8"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">39</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="subsec-Discrete-Random-Variables-8-1-1">
|
||||
<div class="para">Suppose we flip a coin <span class="process-math">\(n\)</span> times. Let <span class="process-math">\(H_i\)</span> indicate heads on flip <span class="process-math">\(i\text{.}\)</span> Let <span class="process-math">\(S\)</span> be the total number of heads in all flips. Then:</div>
|
||||
<div class="displaymath process-math" id="subsec-Discrete-Random-Variables-8-1-1-5">
|
||||
\begin{gather*}
|
||||
S = H_1 + H_2 + \dotsb + H_{n}
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">If <span class="process-math">\(p\)</span> is the probability of the coin coming up heads on a flip, then <span class="process-math">\(S\)</span> has the <dfn class="terminology">binomial distribution</dfn> with parameters <span class="process-math">\(n, p\text{.}\)</span> We’ll use the notation <span class="process-math">\(S \sim \Bin(n, p)\)</span> and:</div>
|
||||
<div class="displaymath process-math" id="subsec-Discrete-Random-Variables-8-1-1-11">
|
||||
\begin{gather*}
|
||||
b(k) = b(k; n, p) = \Pr(S = k).
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-8-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para" id="subsec-Discrete-Random-Variables-8-1-2">Suppose the coin has <span class="process-math">\(p = 0.3\)</span> and we flip it <span class="process-math">\(n = 4\)</span> times. Find <span class="process-math">\(b(2) = b(2; 4, 0.3)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-8-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para logical" id="subsec-Discrete-Random-Variables-8-1-3">
|
||||
<div class="para">The relevant flip sequences are:</div>
|
||||
<div class="displaymath process-math" id="subsec-Discrete-Random-Variables-8-1-3-1">
|
||||
\begin{gather*}
|
||||
THHT, HHTT, TTHH, THTH, HTHT, HTTH
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">Each individual sequence has a probability of <span class="process-math">\((0.3)(0.3)(0.7)(0.7) = 0.0441\text{.}\)</span> So the total probability is:</div>
|
||||
<div class="displaymath process-math" id="subsec-Discrete-Random-Variables-8-1-3-3">
|
||||
\begin{gather*}
|
||||
b(2; 4, 0.3) = (6)(0.0441) = 0.2646.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">That is, (number of flip sequences)(probability of each sequence).</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-8-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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||||
<div class="autopermalink" data-description="Example 39"><a href="#subsec-Discrete-Random-Variables-8" title="Copy heading and permalink for Example 39" aria-label="Copy heading and permalink for Example 39">🔗</a></div></article><div class="autopermalink" data-description="Subsection: Sec 2.1: Random Variables"><a href="#subsec-Discrete-Random-Variables" title="Copy heading and permalink for Subsection: Sec 2.1: Random Variables" aria-label="Copy heading and permalink for Subsection: Sec 2.1: Random Variables">🔗</a></div></section><div class="autopermalink" data-description="Section: Tuesday, Jan 20"><a href="#notes-01-20" title="Copy heading and permalink for Section: Tuesday, Jan 20" aria-label="Copy heading and permalink for Section: Tuesday, Jan 20">🔗</a></div></section></div>
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<div class="toc-title-box"><a href="notes-week-01.html" class="internal"><span class="title">Week 1</span></a></div>
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<div class="toc-title-box"><a href="notes-week-01.html#notes-week-01-3" class="internal"><span class="title">Tuesday 1/13</span></a></div>
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<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Sets" class="internal"><span class="title">Sec 1.1: Sets</span></a></div></li>
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<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Probability" class="internal"><span class="title">Sec 1.2: Probability</span></a></div></li>
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<div class="toc-title-box"><a href="notes-week-01.html#notes-week-01-4" class="internal"><span class="title">Thursday 1/15</span></a></div>
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<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Conditional-Probability" class="internal"><span class="title">Conditional Probability</span></a></div></li>
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<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Diagnostic-Testing" class="internal"><span class="title">Diagnostic Testing</span></a></div></li>
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<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Independent-Events" class="internal"><span class="title">Independent Events</span></a></div></li>
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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="notes-week-01"><h2 class="heading hide-type">
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<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Week 1</span>
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</h2>
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<section class="introduction" id="notes-week-01-2"><div class="para" id="notes-week-01-2-1">This is an outline of the topics we covered in the first week of class. These notes are <em class="emphasis">not</em> a substitute for your own note-taking. I highly recommend that you take your own notes during class. If you ever miss a class for any reason, reach out to another student in class to get a copy of their notes.<div class="autopermalink" data-description="Paragraph"><a href="#notes-week-01-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div></section><section class="subsection" id="notes-week-01-3"><h3 class="heading hide-type">
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<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Tuesday 1/13</span>
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</h3>
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<section class="subsubsection" id="subsubsec-Sets"><h4 class="heading hide-type">
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<span class="type">Subsubsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Sec 1.1: Sets</span>
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</h4>
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<div class="para" id="subsubsec-Sets-2">In prior math courses, you mostly asked <dfn class="terminology">deterministic</dfn> questions. Now, we need new tools to model <dfn class="terminology">randomness</dfn>.<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Sets-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<article class="example example-like" id="subsubsec-Sets-3"><h5 class="heading">
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<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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</h5>
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<div class="para" id="subsubsec-Sets-3-1-1">A toxin molecule in a cell has a certain chance each minute to leave the cell.<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Sets-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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||||
</div>
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||||
<div class="autopermalink" data-description="Example 1"><a href="#subsubsec-Sets-3" title="Copy heading and permalink for Example 1" aria-label="Copy heading and permalink for Example 1">🔗</a></div></article><article class="example example-like" id="subsubsec-Sets-4"><h5 class="heading">
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||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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||||
</h5>
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||||
<div class="para" id="subsubsec-Sets-4-1-1">A patient takes a diagnostic test for a disease and wants to know the chance that they have the disease based on the test result.<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Sets-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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||||
</div>
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||||
<div class="autopermalink" data-description="Example 2"><a href="#subsubsec-Sets-4" title="Copy heading and permalink for Example 2" aria-label="Copy heading and permalink for Example 2">🔗</a></div></article><article class="definition definition-like" id="def-sample-space"><h5 class="heading">
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<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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</h5>
|
||||
<div class="para" id="def-sample-space-1-1">The <dfn class="terminology">sample space</dfn>, often denoted <span class="process-math">\(\Omega\text{,}\)</span> is the set of all possible results of an experiment. A single result is called an <dfn class="terminology">outcome</dfn>, while a collection of results is called an <dfn class="terminology">event</dfn>.<div class="autopermalink" data-description="Paragraph"><a href="#def-sample-space-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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||||
<div class="autopermalink" data-description="Definition 3"><a href="#def-sample-space" title="Copy heading and permalink for Definition 3" aria-label="Copy heading and permalink for Definition 3">🔗</a></div></article><article class="example example-like" id="example-sample-space"><h5 class="heading">
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||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h5>
|
||||
<div class="para" id="example-sample-space-1-1">An experiment consists of rolling a standard 6-sided die (D6). The sample space is <span class="process-math">\(\Omega = \{1, 2, 3, 4, 5, 6\}\text{.}\)</span> One possible event is <span class="process-math">\(A = \{2, 4, 6\}\text{,}\)</span> i.e., the event that the result of the roll is even.<div class="autopermalink" data-description="Paragraph"><a href="#example-sample-space-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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||||
<div class="autopermalink" data-description="Example 4"><a href="#example-sample-space" title="Copy heading and permalink for Example 4" aria-label="Copy heading and permalink for Example 4">🔗</a></div></article><div class="para" id="subsubsec-Sets-7">Note: we’ll use notation like D6 to indicate a 6-sided die with faces 1, 2, 3, 4, 5, 6. Similarly, for example, D4 will indicate a 4-sided die with faces 1, 2, 3, 4.<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Sets-7" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<article class="definition definition-like" id="def-subset"><h5 class="heading">
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<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">5</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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||||
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||||
<div class="para" id="def-subset-1-1">The symbol <span class="process-math">\(\in\)</span> means "is an <dfn class="terminology">element</dfn> of", as in <span class="process-math">\(4 \in A\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-subset-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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||||
</div> <div class="para" id="def-subset-1-2">The symbol <span class="process-math">\(\subset\)</span> means "is a <dfn class="terminology">subset</dfn> of", as in <span class="process-math">\(A \subset \Omega\text{.}\)</span> This means that every element of the set <span class="process-math">\(A\)</span> is also an element of the set <span class="process-math">\(\Omega\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-subset-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Definition 5"><a href="#def-subset" title="Copy heading and permalink for Definition 5" aria-label="Copy heading and permalink for Definition 5">🔗</a></div></article><article class="definition definition-like" id="def-set-operations"><h5 class="heading">
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||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">6</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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||||
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|
||||
<div class="para logical" id="def-set-operations-1-1">
|
||||
<div class="para">Consider sets <span class="process-math">\(A\)</span> and <span class="process-math">\(B\text{,}\)</span> each contained inside <span class="process-math">\(\Omega\text{.}\)</span> We can combine sets in a variety of ways:</div>
|
||||
<dl class="description-list">
|
||||
<dt id="def-set-operations-1-1-4-1">Union<div class="autopermalink" data-description="Item: Union"><a href="#def-set-operations-1-1-4-1" title="Copy heading and permalink for Item: Union" aria-label="Copy heading and permalink for Item: Union">🔗</a></div>
|
||||
</dt>
|
||||
<dd><div class="para" id="def-set-operations-1-1-4-1-2">The <dfn class="terminology">union</dfn> of <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> is the set <span class="process-math">\(A \cup B = \{x \mid x \in A \text{ or } x \in B\}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-set-operations-1-1-4-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></dd>
|
||||
<dt id="def-set-operations-1-1-4-2">Intersection<div class="autopermalink" data-description="Item: Intersection"><a href="#def-set-operations-1-1-4-2" title="Copy heading and permalink for Item: Intersection" aria-label="Copy heading and permalink for Item: Intersection">🔗</a></div>
|
||||
</dt>
|
||||
<dd><div class="para" id="def-set-operations-1-1-4-2-2">The <dfn class="terminology">intersection</dfn> of <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> is the set <span class="process-math">\(A \cap B = \{x \mid x \in A \text{ and } x \in B\}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-set-operations-1-1-4-2-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></dd>
|
||||
<dt id="def-set-operations-1-1-4-3">Difference<div class="autopermalink" data-description="Item: Difference"><a href="#def-set-operations-1-1-4-3" title="Copy heading and permalink for Item: Difference" aria-label="Copy heading and permalink for Item: Difference">🔗</a></div>
|
||||
</dt>
|
||||
<dd><div class="para" id="def-set-operations-1-1-4-3-2">The <dfn class="terminology">set difference</dfn> <span class="process-math">\(A-B\)</span> is the set <span class="process-math">\(A - B = \{x \mid x \in A \text{ and } x \notin B\}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-set-operations-1-1-4-3-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></dd>
|
||||
<dt id="def-set-operations-1-1-4-4">Complement<div class="autopermalink" data-description="Item: Complement"><a href="#def-set-operations-1-1-4-4" title="Copy heading and permalink for Item: Complement" aria-label="Copy heading and permalink for Item: Complement">🔗</a></div>
|
||||
</dt>
|
||||
<dd><div class="para" id="def-set-operations-1-1-4-4-2">The <dfn class="terminology">complement</dfn> of <span class="process-math">\(A\)</span> is the set <span class="process-math">\(A^c = \{x \in \Omega \mid x \notin A\}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-set-operations-1-1-4-4-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></dd>
|
||||
<dt id="def-set-operations-1-1-4-5">Empty Set<div class="autopermalink" data-description="Item: Empty Set"><a href="#def-set-operations-1-1-4-5" title="Copy heading and permalink for Item: Empty Set" aria-label="Copy heading and permalink for Item: Empty Set">🔗</a></div>
|
||||
</dt>
|
||||
<dd><div class="para" id="def-set-operations-1-1-4-5-2">The <dfn class="terminology">empty set</dfn>, usually written <span class="process-math">\(\emptyset\)</span> or <span class="process-math">\(\{\}\text{,}\)</span> is the set which contains no elements.<div class="autopermalink" data-description="Paragraph"><a href="#def-set-operations-1-1-4-5-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></dd>
|
||||
</dl>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#def-set-operations-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 6"><a href="#def-set-operations" title="Copy heading and permalink for Definition 6" aria-label="Copy heading and permalink for Definition 6">🔗</a></div></article><div class="para" id="subsubsec-Sets-10">It’s useful sometimes to draw pictures called <dfn class="terminology">Venn diagrams</dfn> representing set interactions.<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Sets-10" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
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|
||||
<div class="sbsgroup">
|
||||
<div class="sidebyside"><div class="sbsrow" style="margin-left:5%;margin-right:5%;grid-template-columns:44.4444444444444% 44.4444444444444%;column-gap:10%;">
|
||||
<div class="sbspanel sbspanel--top top" style=""><figure class="figure figure-like" id="fig-Venn-diagram-union"><img src="generated/latex-image/fig-Venn-diagram-union-2.svg" role="img" class="contained" alt="described in detail following the image" aria-describedby="fig-Venn-diagram-union-2-description"><details class="image-description" aria-live="polite"><summary title="details"><span class="icon material-symbols-outlined" aria-hidden="true"></span></summary><div id="fig-Venn-diagram-union-2-description"><div class="para" id="fig-Venn-diagram-union-2-1-1">Venn diagram showing sets <span class="process-math">\(A, B\)</span> with the region representing <span class="process-math">\(A \cup B\)</span> shaded.<div class="autopermalink" data-description="Paragraph"><a href="#fig-Venn-diagram-union-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></div></details><figcaption><span class="type">Figure</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">7<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Union<div class="autopermalink" data-description="Figure 7"><a href="#fig-Venn-diagram-union" title="Copy heading and permalink for Figure 7" aria-label="Copy heading and permalink for Figure 7">🔗</a></div></figcaption></figure></div>
|
||||
<div class="sbspanel sbspanel--top top" style=""><figure class="figure figure-like" id="fig-Venn-diagram-intersection"><img src="generated/latex-image/fig-Venn-diagram-intersection-2.svg" role="img" class="contained" alt="described in detail following the image" aria-describedby="fig-Venn-diagram-intersection-2-description"><details class="image-description" aria-live="polite"><summary title="details"><span class="icon material-symbols-outlined" aria-hidden="true"></span></summary><div id="fig-Venn-diagram-intersection-2-description"><div class="para" id="fig-Venn-diagram-intersection-2-1-1">Venn diagram showing sets <span class="process-math">\(A, B\)</span> with the region representing <span class="process-math">\(A \cap B\)</span> shaded.<div class="autopermalink" data-description="Paragraph"><a href="#fig-Venn-diagram-intersection-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></div></details><figcaption><span class="type">Figure</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">8<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Intersection<div class="autopermalink" data-description="Figure 8"><a href="#fig-Venn-diagram-intersection" title="Copy heading and permalink for Figure 8" aria-label="Copy heading and permalink for Figure 8">🔗</a></div></figcaption></figure></div>
|
||||
</div></div>
|
||||
<div class="sidebyside"><div class="sbsrow" style="margin-left:5%;margin-right:5%;grid-template-columns:44.4444444444444% 44.4444444444444%;column-gap:10%;">
|
||||
<div class="sbspanel sbspanel--top top" style=""><figure class="figure figure-like" id="fig-Venn-diagram-difference"><img src="generated/latex-image/fig-Venn-diagram-difference-2.svg" role="img" class="contained" alt="described in detail following the image" aria-describedby="fig-Venn-diagram-difference-2-description"><details class="image-description" aria-live="polite"><summary title="details"><span class="icon material-symbols-outlined" aria-hidden="true"></span></summary><div id="fig-Venn-diagram-difference-2-description"><div class="para" id="fig-Venn-diagram-difference-2-1-1">Venn diagram showing sets <span class="process-math">\(A, B\)</span> with the region representing <span class="process-math">\(A - B\)</span> shaded.<div class="autopermalink" data-description="Paragraph"><a href="#fig-Venn-diagram-difference-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></div></details><figcaption><span class="type">Figure</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">9<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Difference<div class="autopermalink" data-description="Figure 9"><a href="#fig-Venn-diagram-difference" title="Copy heading and permalink for Figure 9" aria-label="Copy heading and permalink for Figure 9">🔗</a></div></figcaption></figure></div>
|
||||
<div class="sbspanel sbspanel--top top" style=""><figure class="figure figure-like" id="fig-Venn-diagram-complement"><img src="generated/latex-image/fig-Venn-diagram-complement-2.svg" role="img" class="contained" alt="described in detail following the image" aria-describedby="fig-Venn-diagram-complement-2-description"><details class="image-description" aria-live="polite"><summary title="details"><span class="icon material-symbols-outlined" aria-hidden="true"></span></summary><div id="fig-Venn-diagram-complement-2-description"><div class="para" id="fig-Venn-diagram-complement-2-1-1">Venn diagram showing set <span class="process-math">\(A \subset \Omega\)</span> with the region representing <span class="process-math">\(A^c\)</span> shaded.<div class="autopermalink" data-description="Paragraph"><a href="#fig-Venn-diagram-complement-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></div></details><figcaption><span class="type">Figure</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">10<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Complement<div class="autopermalink" data-description="Figure 10"><a href="#fig-Venn-diagram-complement" title="Copy heading and permalink for Figure 10" aria-label="Copy heading and permalink for Figure 10">🔗</a></div></figcaption></figure></div>
|
||||
</div></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Subsubsection: Sec 1.1: Sets"><a href="#subsubsec-Sets" title="Copy heading and permalink for Subsubsection: Sec 1.1: Sets" aria-label="Copy heading and permalink for Subsubsection: Sec 1.1: Sets">🔗</a></div></section><section class="subsubsection" id="subsubsec-Probability"><h4 class="heading hide-type">
|
||||
<span class="type">Subsubsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Sec 1.2: Probability</span>
|
||||
</h4>
|
||||
<div class="para" id="subsubsec-Probability-2">Next, we want to start assigning probabilities to each individual outcome so we can then find the probabilities of events.<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Probability-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="example example-like" id="subsubsec-Probability-3"><h5 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">11</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h5>
|
||||
<div class="para" id="subsubsec-Probability-3-1-1">An experiment consists of rolling a D6. The sample space is <span class="process-math">\(\Omega = \{1, 2, 3, 4, 5, 6\}\text{.}\)</span> We might assign probabilities as follows:<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Probability-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <figure class="table table-like" id="subsubsec-Probability-3-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">12<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Distribution for a fair die<div class="autopermalink" data-description="Table 12: Distribution for a fair die"><a href="#subsubsec-Probability-3-1-2" title="Copy heading and permalink for Table 12: Distribution for a fair die" aria-label="Copy heading and permalink for Table 12: Distribution for a fair die">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">1</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">2</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">3</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">4</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">5</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">6</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
</table></div></figure> <div class="para logical" id="subsubsec-Probability-3-1-3">
|
||||
<div class="para">Note that we don’t have to assign the same probability to each outcome. If we do, we call this distribution <dfn class="terminology">uniform</dfn>. If we have some event, such as <span class="process-math">\(A = \{2, 4, 6\}\text{,}\)</span> then we calculate the probability of the event by adding together the probabilities of the outcomes that make up the event:</div>
|
||||
<div class="displaymath process-math" id="subsubsec-Probability-3-1-3-3">
|
||||
\begin{gather*}
|
||||
\Pr(A) = \Pr(2) + \Pr(4) + \Pr(6) = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">Note that this is the same result that we would get if we counted the total number of outcomes in <span class="process-math">\(A\)</span> and divided by the total number of outcomes in <span class="process-math">\(\Omega\text{.}\)</span>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Probability-3-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 11"><a href="#subsubsec-Probability-3" title="Copy heading and permalink for Example 11" aria-label="Copy heading and permalink for Example 11">🔗</a></div></article><article class="definition definition-like" id="def-cardinality"><h5 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">13</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h5>
|
||||
<div class="para" id="def-cardinality-1-1">Given a finite set <span class="process-math">\(A\text{,}\)</span> the <dfn class="terminology">cardinality</dfn> of <span class="process-math">\(A\text{,}\)</span> written <span class="process-math">\(|A|\text{,}\)</span> is the number of elements in <span class="process-math">\(A\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-cardinality-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 13"><a href="#def-cardinality" title="Copy heading and permalink for Definition 13" aria-label="Copy heading and permalink for Definition 13">🔗</a></div></article><article class="fact theorem-like" id="subsubsec-Probability-5"><h5 class="heading">
|
||||
<span class="type">Fact</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">14</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h5>
|
||||
<div class="para logical" id="subsubsec-Probability-5-1-1">
|
||||
<div class="para">If <span class="process-math">\(\Omega\)</span> is a finite probability space with the uniform distribution and <span class="process-math">\(A \subset \Omega\)</span> is an event, then:</div>
|
||||
<div class="displaymath process-math" id="subsubsec-Probability-5-1-1-3">
|
||||
\begin{gather*}
|
||||
\Pr(A) = \frac{|A|}{|\Omega|}
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Probability-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Fact 14"><a href="#subsubsec-Probability-5" title="Copy heading and permalink for Fact 14" aria-label="Copy heading and permalink for Fact 14">🔗</a></div></article><article class="example example-like" id="subsubsec-Probability-6"><h5 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">15</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h5>
|
||||
<div class="para" id="subsubsec-Probability-6-1-1">Suppose we have a weighted die that’s much more likely to come up 6 than any other outcome.<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Probability-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <figure class="table table-like" id="subsubsec-Probability-6-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">16<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Distribution for a fair die<div class="autopermalink" data-description="Table 16: Distribution for a fair die"><a href="#subsubsec-Probability-6-1-2" title="Copy heading and permalink for Table 16: Distribution for a fair die" aria-label="Copy heading and permalink for Table 16: Distribution for a fair die">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">1</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.1</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">2</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.1</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">3</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.1</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">4</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.2</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">5</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.1</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">6</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.4</td>
|
||||
</tr>
|
||||
</table></div></figure> <div class="para logical" id="subsubsec-Probability-6-1-3">
|
||||
<div class="para">With this distribution, if <span class="process-math">\(A = \{2, 4, 6\}\text{,}\)</span> then:</div>
|
||||
<div class="displaymath process-math" id="subsubsec-Probability-6-1-3-2">
|
||||
\begin{gather*}
|
||||
\Pr(A) = 0.1 + 0.2 + 0.4 = 0.7 \neq \frac{3}{6}.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Probability-6-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 15"><a href="#subsubsec-Probability-6" title="Copy heading and permalink for Example 15" aria-label="Copy heading and permalink for Example 15">🔗</a></div></article><article class="example example-like" id="subsubsec-Probability-7"><h5 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">17</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h5>
|
||||
<div class="para" id="subsubsec-Probability-7-1-1">Pick a number from 1 to 10. What is the probability of picking 3?<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Probability-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para logical" id="subsubsec-Probability-7-1-2">
|
||||
<div class="para">Some considerations:</div>
|
||||
<ol class="decimal" id="subsubsec-Probability-7-1-2-1">
|
||||
<li id="subsubsec-Probability-7-1-2-1-1">
|
||||
<div class="para" id="subsubsec-Probability-7-1-2-1-1-1">Are we using the uniform distribution?<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Probability-7-1-2-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Item 1"><a href="#subsubsec-Probability-7-1-2-1-1" title="Copy heading and permalink for Item 1" aria-label="Copy heading and permalink for Item 1">🔗</a></div>
|
||||
</li>
|
||||
<li id="subsubsec-Probability-7-1-2-1-2">
|
||||
<div class="para" id="subsubsec-Probability-7-1-2-1-2-1">What even is the sample space here? Do we need to pick only integers, or is <span class="process-math">\(\pi\)</span> an outcome here?<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Probability-7-1-2-1-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Item 2"><a href="#subsubsec-Probability-7-1-2-1-2" title="Copy heading and permalink for Item 2" aria-label="Copy heading and permalink for Item 2">🔗</a></div>
|
||||
</li>
|
||||
<li id="subsubsec-Probability-7-1-2-1-3">
|
||||
<div class="para" id="subsubsec-Probability-7-1-2-1-3-1">What would "uniform" mean if the sample space was infinite?<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Probability-7-1-2-1-3-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Item 3"><a href="#subsubsec-Probability-7-1-2-1-3" title="Copy heading and permalink for Item 3" aria-label="Copy heading and permalink for Item 3">🔗</a></div>
|
||||
</li>
|
||||
</ol>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Probability-7-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 17"><a href="#subsubsec-Probability-7" title="Copy heading and permalink for Example 17" aria-label="Copy heading and permalink for Example 17">🔗</a></div></article><article class="definition definition-like" id="def-probability-distribution"><h5 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">18</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h5>
|
||||
<div class="para logical" id="def-probability-distribution-1-1">
|
||||
<div class="para">A <dfn class="terminology">probability distribution</dfn> on a sample space <span class="process-math">\(\Omega\)</span> assigns probabilities to every event, satisfying the following conditions:</div>
|
||||
<ol class="decimal" id="def-probability-distribution-1-1-3">
|
||||
<li id="def-probability-distribution-1-1-3-1">
|
||||
<div class="para" id="def-probability-distribution-1-1-3-1-1">
|
||||
<span class="process-math">\(\Pr(\Omega) = 1\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-probability-distribution-1-1-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Item 1"><a href="#def-probability-distribution-1-1-3-1" title="Copy heading and permalink for Item 1" aria-label="Copy heading and permalink for Item 1">🔗</a></div>
|
||||
</li>
|
||||
<li id="def-probability-distribution-1-1-3-2">
|
||||
<div class="para" id="def-probability-distribution-1-1-3-2-1">
|
||||
<span class="process-math">\(0 \leq \Pr(A) \leq 1\)</span> for any event <span class="process-math">\(A\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-probability-distribution-1-1-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Item 2"><a href="#def-probability-distribution-1-1-3-2" title="Copy heading and permalink for Item 2" aria-label="Copy heading and permalink for Item 2">🔗</a></div>
|
||||
</li>
|
||||
<li id="def-probability-distribution-1-1-3-3">
|
||||
<div class="para" id="def-probability-distribution-1-1-3-3-1">If <span class="process-math">\(A \cap B = \emptyset\text{,}\)</span> then <span class="process-math">\(\Pr(A\cup B) = \Pr(A) + \Pr(B)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-probability-distribution-1-1-3-3-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Item 3"><a href="#def-probability-distribution-1-1-3-3" title="Copy heading and permalink for Item 3" aria-label="Copy heading and permalink for Item 3">🔗</a></div>
|
||||
</li>
|
||||
</ol>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#def-probability-distribution-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 18"><a href="#def-probability-distribution" title="Copy heading and permalink for Definition 18" aria-label="Copy heading and permalink for Definition 18">🔗</a></div></article><article class="example example-like" id="subsubsec-Probability-9"><h5 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">19</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h5>
|
||||
<div class="para" id="subsubsec-Probability-9-1-1">Suppose we flip a coin until we see heads. The sample space is <span class="process-math">\(\Omega = \{H, TH, TTH, TTTH, \dotsc\}\text{.}\)</span> What would "uniform" mean here? Is there any way we could assign the same probability to each individual outcome here?<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Probability-9-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para" id="subsubsec-Probability-9-1-2">Instead of "uniform", what if we want to treat the coin as a fair coin? Then what would the distribution be?<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Probability-9-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <figure class="table table-like" id="subsubsec-Probability-9-1-3"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">20<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Distribution assuming a fair coin<div class="autopermalink" data-description="Table 20: Distribution assuming a fair coin"><a href="#subsubsec-Probability-9-1-3" title="Copy heading and permalink for Table 20: Distribution assuming a fair coin" aria-label="Copy heading and permalink for Table 20: Distribution assuming a fair coin">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"># flips</td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(H\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines">1</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/2\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(TH\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines">2</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/4\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(TTH\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines">3</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/8\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(TTTH\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines">4</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/16\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(T\dotsm TH\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(n\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/2^n\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
|
||||
</tr>
|
||||
</table></div></figure><div class="autopermalink" data-description="Example 19"><a href="#subsubsec-Probability-9" title="Copy heading and permalink for Example 19" aria-label="Copy heading and permalink for Example 19">🔗</a></div></article><article class="example example-like" id="subsubsec-Probability-10"><h5 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">21</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h5>
|
||||
<div class="para" id="subsubsec-Probability-10-1-1">One last (very useful!) observation. Recall: If <span class="process-math">\(A\cap B = \emptyset\text{,}\)</span> then <span class="process-math">\(\Pr(A\cup B) = \Pr(A) + \Pr(B)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Probability-10-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para logical" id="subsubsec-Probability-10-1-2">
|
||||
<div class="para">For any event <span class="process-math">\(A\text{,}\)</span> <span class="process-math">\(A\cap A^c = \emptyset\)</span> and <span class="process-math">\(A \cup A^c = \Omega\text{.}\)</span> So:</div>
|
||||
<div class="displaymath process-math" id="subsubsec-Probability-10-1-2-4">
|
||||
\begin{gather*}
|
||||
1 = \Pr(\underbrace{A \cup A^c}_{\Omega}) = \Pr(A) + \Pr(A^c).
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">We can rewrite this in two useful ways:</div>
|
||||
<div class="displaymath process-math" id="subsubsec-Probability-10-1-2-5">
|
||||
\begin{align*}
|
||||
\Pr(A) \amp = 1 - \Pr(A^c) \\
|
||||
\Pr(A^c) \amp = 1 - \Pr(A)
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Probability-10-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 21"><a href="#subsubsec-Probability-10" title="Copy heading and permalink for Example 21" aria-label="Copy heading and permalink for Example 21">🔗</a></div></article><div class="autopermalink" data-description="Subsubsection: Sec 1.2: Probability"><a href="#subsubsec-Probability" title="Copy heading and permalink for Subsubsection: Sec 1.2: Probability" aria-label="Copy heading and permalink for Subsubsection: Sec 1.2: Probability">🔗</a></div></section><div class="autopermalink" data-description="Subsection: Tuesday 1/13"><a href="#notes-week-01-3" title="Copy heading and permalink for Subsection: Tuesday 1/13" aria-label="Copy heading and permalink for Subsection: Tuesday 1/13">🔗</a></div></section><section class="subsection" id="notes-week-01-4"><h3 class="heading hide-type">
|
||||
<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Thursday 1/15</span>
|
||||
</h3>
|
||||
<section class="subsubsection" id="subsubsec-Conditional-Probability"><h4 class="heading hide-type">
|
||||
<span class="type">Subsubsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Conditional Probability</span>
|
||||
</h4>
|
||||
<div class="para" id="subsubsec-Conditional-Probability-2">Question: How does evidence (e.g., knowledge of one event occurring) change our knowledge of probabilities for other events?<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Conditional-Probability-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="example example-like" id="subsubsec-Conditional-Probability-3"><h5 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">22</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h5>
|
||||
<div class="para" id="subsubsec-Conditional-Probability-3-1-1">Roll a fair D6 two times. Let <span class="process-math">\(A = \{\text{sum } \geq 10\}\)</span> and <span class="process-math">\(B = \{\text{first roll is } 6\}\text{.}\)</span> <span class="process-math">\(A\)</span> feels more likely if we already know <span class="process-math">\(B\)</span> has occurred.<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Conditional-Probability-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 22"><a href="#subsubsec-Conditional-Probability-3" title="Copy heading and permalink for Example 22" aria-label="Copy heading and permalink for Example 22">🔗</a></div></article><article class="definition definition-like" id="def-conditional-probability"><h5 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">23</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h5>
|
||||
<div class="para logical" id="def-conditional-probability-1-1">
|
||||
<div class="para">The <dfn class="terminology">conditional probability</dfn> of <span class="process-math">\(A\)</span> given <span class="process-math">\(B\)</span> is: ,</div>
|
||||
<div class="displaymath process-math" id="def-conditional-probability-1-1-4">
|
||||
\begin{gather*}
|
||||
\Pr(A \mid B) = \frac{\Pr(A\cap B)}{\Pr(B)}
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#def-conditional-probability-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <figure class="figure figure-like" id="fig-conditional-probability"><div class="image-box" style="width: 50%; margin-left: 25%; margin-right: 25%;">
|
||||
<img src="generated/latex-image/fig-conditional-probability-2.svg" role="img" class="contained" alt="described in detail following the image" aria-describedby="fig-conditional-probability-2-description"><details class="image-description" aria-live="polite"><summary title="details"><span class="icon material-symbols-outlined" aria-hidden="true"></span></summary><div id="fig-conditional-probability-2-description"><div class="para" id="fig-conditional-probability-2-1-1">Two overlapping circles representing events <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> sit inside a rectangle representing the sample space <span class="process-math">\(\Omega\text{.}\)</span> The circle labeled <span class="process-math">\(B\)</span> is shaded. The portion of that circle which is overlapped by the <span class="process-math">\(A\)</span> circle is also filled in with slanted lines.<div class="autopermalink" data-description="Paragraph"><a href="#fig-conditional-probability-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></div></details>
|
||||
</div>
|
||||
<figcaption><span class="type">Figure</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">24<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="process-math">\(\Pr(A \mid B)\)</span> tells the proportion of <span class="process-math">\(B\)</span> which is overlapped by <span class="process-math">\(A\text{.}\)</span><div class="autopermalink" data-description="Figure 24"><a href="#fig-conditional-probability" title="Copy heading and permalink for Figure 24" aria-label="Copy heading and permalink for Figure 24">🔗</a></div></figcaption></figure><div class="autopermalink" data-description="Definition 23"><a href="#def-conditional-probability" title="Copy heading and permalink for Definition 23" aria-label="Copy heading and permalink for Definition 23">🔗</a></div></article><article class="example example-like" id="example-rolls-conditional"><h5 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">25</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h5>
|
||||
<div class="para logical" id="example-rolls-conditional-1-1">
|
||||
<div class="para">Continuing from the previous example, <span class="process-math">\(|\Omega| = 36\text{.}\)</span>
|
||||
</div>
|
||||
<div class="displaymath process-math" id="example-rolls-conditional-1-1-2">
|
||||
\begin{align*}
|
||||
A \amp = \{(4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)\} \\
|
||||
B \amp = \{(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\} \\
|
||||
A \cap B \amp = \{(6, 4), (6, 5), (6, 6)\}
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">So <span class="process-math">\(\Pr(A) = \frac{6}{36}, \Pr(B) = \frac{6}{36}, \text{and } \Pr(A\cap B) = \frac{3}{36}\text{.}\)</span> Then:</div>
|
||||
<div class="displaymath process-math" id="example-rolls-conditional-1-1-4">
|
||||
\begin{gather*}
|
||||
\Pr(A \mid B) = \frac{\Pr(A\cap B)}{\Pr(B)} = \frac{3/36}{6/36} = \frac{3}{6} = \frac{1}{2}.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">Notice that <span class="process-math">\(\Pr(A\mid B)\)</span> is significantly larger than <span class="process-math">\(\Pr(A)\text{.}\)</span>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#example-rolls-conditional-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 25"><a href="#example-rolls-conditional" title="Copy heading and permalink for Example 25" aria-label="Copy heading and permalink for Example 25">🔗</a></div></article><div class="autopermalink" data-description="Subsubsection: Conditional Probability"><a href="#subsubsec-Conditional-Probability" title="Copy heading and permalink for Subsubsection: Conditional Probability" aria-label="Copy heading and permalink for Subsubsection: Conditional Probability">🔗</a></div></section><section class="subsubsection" id="subsubsec-Diagnostic-Testing"><h4 class="heading hide-type">
|
||||
<span class="type">Subsubsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Diagnostic Testing</span>
|
||||
</h4>
|
||||
<div class="para" id="subsubsec-Diagnostic-Testing-2">Setup: A patient takes a diagnostic test. Let <span class="process-math">\(P\)</span> be the event that they test positive. Let <span class="process-math">\(D\)</span> be the event that they have the disease.<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Diagnostic-Testing-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="definition definition-like" id="def-sensitivity-specificity"><h5 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">26</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h5>
|
||||
<div class="para" id="def-sensitivity-specificity-1-1">The <dfn class="terminology">sensitivity</dfn> of a diagnostic test is <span class="process-math">\(\Pr(P \mid D)\text{.}\)</span> The <dfn class="terminology">specificity</dfn> of a diagnostic test is <span class="process-math">\(\Pr(P^c \mid D^c)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-sensitivity-specificity-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 26"><a href="#def-sensitivity-specificity" title="Copy heading and permalink for Definition 26" aria-label="Copy heading and permalink for Definition 26">🔗</a></div></article><div class="para" id="subsubsec-Diagnostic-Testing-4">But, what the patient really wants to know is <span class="process-math">\(\Pr(D \mid P)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Diagnostic-Testing-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="example example-like" id="subsubsec-Diagnostic-Testing-5"><h5 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">27</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h5>
|
||||
<div class="para" id="subsubsec-Diagnostic-Testing-5-1-1">A disease has a prevalence of 1%. A test has sensitivity of 90% and specificity of 91%. For a patient who gets a positive test result, what is the probability that they have the disease?<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Diagnostic-Testing-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para logical" id="subsubsec-Diagnostic-Testing-5-1-2">
|
||||
<div class="displaymath process-math" id="subsubsec-Diagnostic-Testing-5-1-2-1">
|
||||
\begin{align*}
|
||||
\text{A) } 9/10 \amp \amp \text{B) } 8/10 \amp \amp \text{C) } 1/10 \amp \amp \text{D) } 1/100
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Diagnostic-Testing-5-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="subsubsec-Diagnostic-Testing-5-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
|
||||
<div class="para" id="subsubsec-Diagnostic-Testing-5-2-1">C!<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Diagnostic-Testing-5-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Answer 27.1"><a href="#subsubsec-Diagnostic-Testing-5-2" title="Copy heading and permalink for Answer 27.1" aria-label="Copy heading and permalink for Answer 27.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Example 27"><a href="#subsubsec-Diagnostic-Testing-5" title="Copy heading and permalink for Example 27" aria-label="Copy heading and permalink for Example 27">🔗</a></div></article><article class="theorem theorem-like" id="thm-Bayes-v1"><h5 class="heading">
|
||||
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">28</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Bayes’ Theorem (v1).</span>
|
||||
</h5>
|
||||
<div class="para logical" id="thm-Bayes-v1-2-1">
|
||||
<div class="para">For events <span class="process-math">\(A, B\)</span> with nonzero probability:</div>
|
||||
<div class="displaymath process-math" id="thm-Bayes-v1-2-1-2">
|
||||
\begin{gather*}
|
||||
\Pr(B \mid A) = \frac{\Pr(A \mid B)\Pr(B)}{\Pr(A)}
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#thm-Bayes-v1-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Theorem 28: Bayes’ Theorem (v1)"><a href="#thm-Bayes-v1" title="Copy heading and permalink for Theorem 28: Bayes’ Theorem (v1)" aria-label="Copy heading and permalink for Theorem 28: Bayes’ Theorem (v1)">🔗</a></div></article><div class="para logical" id="subsubsec-Diagnostic-Testing-7">
|
||||
<div class="para">For example, if a patient sees a positive diagnostic test result, they might try to calculate:</div>
|
||||
<div class="displaymath process-math" id="subsubsec-Diagnostic-Testing-7-1">
|
||||
\begin{gather*}
|
||||
\Pr(D \mid P) = \frac{\Pr(P \mid D)\Pr(D)}{\Pr(P)}
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">
|
||||
<span class="process-math">\(\Pr(P \mid D)\)</span> is the sensitivity. <span class="process-math">\(\Pr(D)\)</span> could be the prevalence. We don’t have direct access to <span class="process-math">\(\Pr(P)\text{.}\)</span>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Diagnostic-Testing-7" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="para" id="subsubsec-Diagnostic-Testing-8">Observation: <span class="process-math">\(\Omega = D \cup D^c\text{,}\)</span> so <span class="process-math">\(P = (P\cap D) \cup (P\cap D^c)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Diagnostic-Testing-8" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<figure class="figure figure-like" id="fig-P-breakdown"><div class="image-box" style="width: 50%; margin-left: 25%; margin-right: 25%;">
|
||||
<img src="generated/latex-image/fig-P-breakdown-2.svg" role="img" class="contained" alt="described in detail following the image" aria-describedby="fig-P-breakdown-2-description"><details class="image-description" aria-live="polite"><summary title="details"><span class="icon material-symbols-outlined" aria-hidden="true"></span></summary><div id="fig-P-breakdown-2-description"><div class="para" id="fig-P-breakdown-2-1-1"><div class="autopermalink" data-description="Paragraph"><a href="#fig-P-breakdown-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div></div></div></details>
|
||||
</div>
|
||||
<figcaption><span class="type">Figure</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">29<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" data-description="Figure 29"><a href="#fig-P-breakdown" title="Copy heading and permalink for Figure 29" aria-label="Copy heading and permalink for Figure 29">🔗</a></div></figcaption></figure><div class="para logical" id="subsubsec-Diagnostic-Testing-10">
|
||||
<div class="para">Observation 2:</div>
|
||||
<div class="displaymath process-math" id="subsubsec-Diagnostic-Testing-10-1">
|
||||
\begin{align*}
|
||||
\Pr(P \mid D) \amp \frac{\Pr(P\cap D)}{\Pr(D)} \amp \amp \Rightarrow \amp \Pr(P \cap D) \amp = \Pr(P\mid D)\Pr(D) \\
|
||||
\Pr(P \mid D^c) \amp \frac{\Pr(P\cap D^c)}{\Pr(D^c)} \amp \amp \Rightarrow \amp \Pr(P \cap D^c) \amp = \Pr(P\mid D^c)\Pr(D^c)
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">So:</div>
|
||||
<div class="displaymath process-math" id="subsubsec-Diagnostic-Testing-10-2">
|
||||
\begin{gather*}
|
||||
\Pr(P) = \Pr(P\mid D)\Pr(D) + \Pr(P\mid D^c)\Pr(D^c)
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Diagnostic-Testing-10" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="theorem theorem-like" id="thm-Bayes-v2"><h5 class="heading">
|
||||
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">30</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Bayes’ Theorem (v2).</span>
|
||||
</h5>
|
||||
<div class="para logical" id="thm-Bayes-v2-2-1">
|
||||
<div class="displaymath process-math" id="thm-Bayes-v2-2-1-1">
|
||||
\begin{gather*}
|
||||
\Pr(B \mid A) = \frac{\Pr(A \mid B)\Pr(B)}{\Pr(A \mid B)\Pr(B) + \Pr(A \mid B^c)\Pr(B^c)}
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#thm-Bayes-v2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Theorem 30: Bayes’ Theorem (v2)"><a href="#thm-Bayes-v2" title="Copy heading and permalink for Theorem 30: Bayes’ Theorem (v2)" aria-label="Copy heading and permalink for Theorem 30: Bayes’ Theorem (v2)">🔗</a></div></article><article class="example example-like" id="subsubsec-Diagnostic-Testing-12"><h5 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">31</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h5>
|
||||
<div class="para logical" id="subsubsec-Diagnostic-Testing-12-1-1">
|
||||
<div class="para">Continuing from the previous example:</div>
|
||||
<div class="displaymath process-math" id="subsubsec-Diagnostic-Testing-12-1-1-1">
|
||||
\begin{align*}
|
||||
\Pr(D\mid P) \amp = \frac{\Pr(P\mid D)\Pr(D)}{\Pr(P\mid D)\Pr(D) + \Pr(P\mid D^c)\Pr(D^c)} \\
|
||||
\amp = \frac{(0.9)(0.01)}{(0.9)(0.01) + (1 - 0.91)(1 - 0.01)} \\
|
||||
\amp \approx 0.092
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">What if the patient got a negative test result instead? In that case, what is the probaiblity they do not have the disease?</div>
|
||||
<div class="displaymath process-math" id="subsubsec-Diagnostic-Testing-12-1-1-2">
|
||||
\begin{align*}
|
||||
\Pr(D^c\mid P^c) \amp = \frac{\Pr(P^c\mid D^c)\Pr(D^c)}{\Pr(P^c\mid D^c)\Pr(D^c) + \Pr(P^c\mid D)\Pr(D)} \\
|
||||
\amp = \frac{(0.91)(1 - 0.01)}{(0.91)(1 - 0.01) + (1 - 0.9)(0.01)} \\
|
||||
\amp \approx 0.999
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Diagnostic-Testing-12-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 31"><a href="#subsubsec-Diagnostic-Testing-12" title="Copy heading and permalink for Example 31" aria-label="Copy heading and permalink for Example 31">🔗</a></div></article><div class="autopermalink" data-description="Subsubsection: Diagnostic Testing"><a href="#subsubsec-Diagnostic-Testing" title="Copy heading and permalink for Subsubsection: Diagnostic Testing" aria-label="Copy heading and permalink for Subsubsection: Diagnostic Testing">🔗</a></div></section><section class="subsubsection" id="subsubsec-Independent-Events"><h4 class="heading hide-type">
|
||||
<span class="type">Subsubsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Independent Events</span>
|
||||
</h4>
|
||||
<div class="para" id="subsubsec-Independent-Events-2">Question: <span class="process-math">\(\Pr(A \mid B)\)</span> is supposed to capture how information about <span class="process-math">\(B\)</span> affects the probability of <span class="process-math">\(A\text{.}\)</span> What if it doesn’t?<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Independent-Events-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="definition definition-like" id="def-independent-events"><h5 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">32</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h5>
|
||||
<div class="para" id="def-independent-events-1-1">Events <span class="process-math">\(A, B\)</span> are <dfn class="terminology">independent</dfn> if <span class="process-math">\(\Pr(A \mid B) = \Pr(A)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-independent-events-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 32"><a href="#def-independent-events" title="Copy heading and permalink for Definition 32" aria-label="Copy heading and permalink for Definition 32">🔗</a></div></article><div class="para logical" id="subsubsec-Independent-Events-4">
|
||||
<div class="para">Observation: If <span class="process-math">\(A, B\)</span> have nonzero probability and are independent, then:</div>
|
||||
<div class="displaymath process-math" id="subsubsec-Independent-Events-4-2">
|
||||
\begin{align*}
|
||||
\Pr(A\mid B) \amp = \Pr(A) \\
|
||||
\frac{\Pr(A\cap B)}{\Pr(B)} \amp = \Pr(A) \\
|
||||
\Pr(A\cap B) \amp = \Pr(A)\Pr(B)
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">We can take this last equation as a definition of independence.</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsubsec-Independent-Events-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="example example-like" id="example-rolls-independent"><h5 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">33</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h5>
|
||||
<div class="para" id="example-rolls-independent-1-1">Continuing <a href="notes-week-01.html#example-rolls-conditional" class="xref" data-knowl="./knowl/xref/example-rolls-conditional.html" data-reveal-label="Reveal" data-close-label="Close" title="Example 25">Example 25</a>, recall <span class="process-math">\(A = \{\text{sum} \geq 10\}\)</span> and <span class="process-math">\(B = \{\text{1st roll is } 6\}\text{.}\)</span> We found that <span class="process-math">\(\Pr(A \mid B) \neq \Pr(A)\text{,}\)</span> so <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> are not independent.<div class="autopermalink" data-description="Paragraph"><a href="#example-rolls-independent-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para logical" id="example-rolls-independent-1-2">
|
||||
<div class="para">Now consider the event <span class="process-math">\(C = \{\text{sum} = 7\}\text{.}\)</span> We have:</div>
|
||||
<div class="displaymath process-math" id="example-rolls-independent-1-2-2">
|
||||
\begin{align*}
|
||||
C \amp = \{(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)\} \\
|
||||
\Pr(C) \amp = \frac{6}{36} = \frac{1}{6} \\
|
||||
B\cap C \amp = \{(6, 1)\} \\
|
||||
\Pr(B\cap C) \amp = \frac{1}{36} \\
|
||||
\text{therefore: } \Pr(C\mid B) \amp = \frac{\Pr(C\cap B)}{\Pr(B)} = \frac{1/36}{1/6} = \frac{1}{6} = \Pr(C)
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">Therefore events <span class="process-math">\(B, C\)</span> are independent.</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#example-rolls-independent-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 33"><a href="#example-rolls-independent" title="Copy heading and permalink for Example 33" aria-label="Copy heading and permalink for Example 33">🔗</a></div></article><div class="autopermalink" data-description="Subsubsection: Independent Events"><a href="#subsubsec-Independent-Events" title="Copy heading and permalink for Subsubsection: Independent Events" aria-label="Copy heading and permalink for Subsubsection: Independent Events">🔗</a></div></section><div class="autopermalink" data-description="Subsection: Thursday 1/15"><a href="#notes-week-01-4" title="Copy heading and permalink for Subsection: Thursday 1/15" aria-label="Copy heading and permalink for Subsection: Thursday 1/15">🔗</a></div></section><div class="autopermalink" data-description="Section: Week 1"><a href="#notes-week-01" title="Copy heading and permalink for Section: Week 1" aria-label="Copy heading and permalink for Section: Week 1">🔗</a></div></section></div>
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<h1 class="heading"><a href="root-1-2.html"><span class="title">Introduction to Probability and Statistics for the Life Sciences</span> <span class="subtitle">Spring 2026 MATH 1044</span></a></h1>
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<p class="byline">Andy Eisenberg</p>
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<section class="worksheet" id="quiz-01" data-margins="0.75in 0.75in 0.75in 0.75in"><h1 class="heading hide-type">
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<span class="type">Worksheet</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Quiz 1</span><div class="print-links"><a href="quiz-01-printable.html" class="print-link" title="Print preview"><span class="icon material-symbols-outlined" aria-hidden="true"></span></a></div>
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</h1>
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<section class="introduction" id="quiz-01-2"><div class="para" id="quiz-01-2-1">The following work should be completed individually. Use of notes or textbooks is not allowed. You may use a scientific calculator, not a graphing calculator or phone app.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="para" id="quiz-01-2-2">Show all work unless instructed otherwise.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-2-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div></section><section class="onepage" id="quiz-01-3"><article class="exercise exercise-like" id="quiz-01-3-1"><h3 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
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<div class="introduction" id="quiz-01-3-1-1"><div class="para" id="quiz-01-3-1-1-1">Write either True or False for each of the following statements. No justification is required.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div></div>
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<article class="task exercise-like" id="quiz-01-3-1-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
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<div class="para" id="quiz-01-3-1-2-1-1">If <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> are any sets, then <span class="process-math">\(|A\cap B| \leq |A|\)</span> and <span class="process-math">\(|A\cap B| \leq |B|\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-1-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="solutions"><details id="quiz-01-3-1-2-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
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<div class="para" id="quiz-01-3-1-2-2-1">True.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-1-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Solution 1.a.1"><a href="#quiz-01-3-1-2-2" title="Copy heading and permalink for Solution 1.a.1" aria-label="Copy heading and permalink for Solution 1.a.1">🔗</a></div>
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</div></details></div>
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<div class="autopermalink" data-description="Task 1.a"><a href="#quiz-01-3-1-2" title="Copy heading and permalink for Task 1.a" aria-label="Copy heading and permalink for Task 1.a">🔗</a></div></article><article class="task exercise-like" id="quiz-01-3-1-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
|
||||
<div class="para" id="quiz-01-3-1-3-1-1">If <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> are any events, then <span class="process-math">\(\Pr(A \mid B) = 1 - \Pr(B \mid A)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-1-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="solutions"><details id="quiz-01-3-1-3-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
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||||
<div class="para" id="quiz-01-3-1-3-2-1">False.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-1-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Solution 1.b.1"><a href="#quiz-01-3-1-3-2" title="Copy heading and permalink for Solution 1.b.1" aria-label="Copy heading and permalink for Solution 1.b.1">🔗</a></div>
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</div></details></div>
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<div class="autopermalink" data-description="Task 1.b"><a href="#quiz-01-3-1-3" title="Copy heading and permalink for Task 1.b" aria-label="Copy heading and permalink for Task 1.b">🔗</a></div></article><article class="task exercise-like" id="quiz-01-3-1-4"><h4 class="heading"><span class="codenumber">(c)</span></h4>
|
||||
<div class="para" id="quiz-01-3-1-4-1-1">If <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> are any events, then <span class="process-math">\(\Pr(A\cap B)\Pr(B) = \Pr(A \mid B)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-1-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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||||
<div class="solutions"><details id="quiz-01-3-1-4-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
|
||||
<div class="para" id="quiz-01-3-1-4-2-1">False.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-1-4-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Solution 1.c.1"><a href="#quiz-01-3-1-4-2" title="Copy heading and permalink for Solution 1.c.1" aria-label="Copy heading and permalink for Solution 1.c.1">🔗</a></div>
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</div></details></div>
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<div class="autopermalink" data-description="Task 1.c"><a href="#quiz-01-3-1-4" title="Copy heading and permalink for Task 1.c" aria-label="Copy heading and permalink for Task 1.c">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 1"><a href="#quiz-01-3-1" title="Copy heading and permalink for Worksheet Exercise 1" aria-label="Copy heading and permalink for Worksheet Exercise 1">🔗</a></div></article><article class="exercise exercise-like" id="quiz-01-3-2"><h3 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<div class="introduction" id="quiz-01-3-2-1"><div class="para" id="quiz-01-3-2-1-1">We find a 4-sided die with faces 0, 1, 3, and 5. An experiment consists of rolling the die two times.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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||||
</div></div>
|
||||
<article class="task exercise-like" id="quiz-01-3-2-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
|
||||
<div class="para" id="quiz-01-3-2-2-1-1">Write down the sample space <span class="process-math">\(\Omega\)</span> of all possible outcomes for this experiment.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-2-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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||||
<div class="solutions"><details id="quiz-01-3-2-2-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
|
||||
<div class="para logical" id="quiz-01-3-2-2-2-1">
|
||||
<div class="displaymath process-math" id="quiz-01-3-2-2-2-1-1">
|
||||
\begin{align*}
|
||||
\Omega = \{ \amp (0, 0), (0, 1), (0, 3), (0, 5), \\
|
||||
\amp (1, 0), (1, 1), (1, 3), (1, 5), \\
|
||||
\amp (3, 0), (3, 1), (3, 3), (3, 5), \\
|
||||
\amp (5, 0), (5, 1), (5, 3), (5, 5)\}
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-2-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Solution 2.a.1"><a href="#quiz-01-3-2-2-2" title="Copy heading and permalink for Solution 2.a.1" aria-label="Copy heading and permalink for Solution 2.a.1">🔗</a></div>
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||||
</div></details></div>
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<div class="autopermalink" data-description="Task 2.a"><a href="#quiz-01-3-2-2" title="Copy heading and permalink for Task 2.a" aria-label="Copy heading and permalink for Task 2.a">🔗</a></div></article><article class="task exercise-like" id="quiz-01-3-2-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
|
||||
<div class="para" id="quiz-01-3-2-3-1-1">Let <span class="process-math">\(A\)</span> be the event that the second roll is at least twice the value of the first roll. Let m B be the event that the sum of the rolls is odd. Assume the die is fair. List the outcomes in <span class="process-math">\(A\)</span> and calculate <span class="process-math">\(\Pr(A)\text{.}\)</span> List the outcomes in <span class="process-math">\(B\)</span> and calculate <span class="process-math">\(\Pr(B)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-2-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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||||
</div>
|
||||
<div class="solutions"><details id="quiz-01-3-2-3-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
|
||||
<div class="para logical" id="quiz-01-3-2-3-2-1">
|
||||
<div class="displaymath process-math" id="quiz-01-3-2-3-2-1-1">
|
||||
\begin{align*}
|
||||
A \amp = \{(0, 0), (0, 1), (0, 3), (0, 5), (1, 3), (1, 5)\} \\
|
||||
B \amp = \{ (0, 1), (0, 3), (0, 5), (1, 0), (3, 0), (5, 0) \}
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">Then <span class="process-math">\(\Pr(A) = \frac{|A|}{|\Omega|} = \frac{6}{16} = \frac{3}{8}\text{,}\)</span> and <span class="process-math">\(\Pr(B) = \frac{|B|}{|\Omega|} = \frac{6}{16} = \frac{3}{8}\text{.}\)</span>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-2-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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||||
</div>
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<div class="para" id="quiz-01-4-1-1-1">A diagnostic test is developed to detect a disease present in 1.3% of the population. For a patient who has the disease, the test will accurately give a positive result 62% of the time. When the patient does not have the disease, the test will accurately give a negative result 99.4% of the time.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-4-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div> <div class="para" id="quiz-01-4-1-1-2">For a patient who receives a positive test, what is the probability they have the disease?<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-4-1-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="para logical" id="quiz-01-4-1-2-1">
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<div class="para">Let <span class="process-math">\(P\)</span> be the event of receiving a positive test result and <span class="process-math">\(D\)</span> be the event of having the disease. The given information is: <span class="process-math">\(\Pr(D) = 0.013, \Pr(P\mid D) = 0.62\text{,}\)</span> and <span class="process-math">\(\Pr(P^c \mid D^c) = 0.994\text{.}\)</span> Then, using Bayes’ Theorem:</div>
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<div class="displaymath process-math" id="quiz-01-4-1-2-1-5">
|
||||
\begin{align*}
|
||||
\Pr(D\mid P) = \frac{\Pr(P \mid D)\Pr(D)}{\Pr(P)} \amp = \frac{\Pr(P \mid D)\Pr(D)}{\Pr(P \mid D)\Pr(D) + \Pr(P \mid D^c)\Pr(D^c)} \\
|
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\amp = \frac{\Pr(P \mid D)\Pr(D)}{\Pr(P \mid D)\Pr(D) + (1 - \Pr(P^c \mid D^c))(1 - \Pr(D))} \\
|
||||
\amp = \frac{(0.62)(0.013)}{(0.62)(0.013) + (1 - 0.994)(1 - 0.013)} \\
|
||||
\amp \approx \boxed{0.576}
|
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\end{align*}
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<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-20.html#subsec-Discrete-Random-Variables" class="internal"><span class="title">Sec 2.1: Random Variables</span></a></div></li></ul>
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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="worksheet" id="quiz-01" data-margins="0.75in 0.75in 0.75in 0.75in"><h2 class="heading hide-type">
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<span class="type">Worksheet</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Quiz 1</span><div class="print-links"><a href="quiz-01-printable.html" class="print-link" title="Print preview"><span class="icon material-symbols-outlined" aria-hidden="true"></span></a></div>
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</h2>
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<section class="introduction" id="quiz-01-2"><div class="para" id="quiz-01-2-1">The following work should be completed individually. Use of notes or textbooks is not allowed. You may use a scientific calculator, not a graphing calculator or phone app.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="para" id="quiz-01-2-2">Show all work unless instructed otherwise.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-2-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div></section><section class="onepage" id="quiz-01-3"><article class="exercise exercise-like" id="quiz-01-3-1"><h3 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
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<div class="introduction" id="quiz-01-3-1-1"><div class="para" id="quiz-01-3-1-1-1">Write either True or False for each of the following statements. No justification is required.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<article class="task exercise-like" id="quiz-01-3-1-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
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<div class="para" id="quiz-01-3-1-2-1-1">If <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> are any sets, then <span class="process-math">\(|A\cap B| \leq |A|\)</span> and <span class="process-math">\(|A\cap B| \leq |B|\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-1-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="solutions"><details id="quiz-01-3-1-2-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
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<div class="para" id="quiz-01-3-1-2-2-1">True.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-1-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Solution 1.a.1"><a href="#quiz-01-3-1-2-2" title="Copy heading and permalink for Solution 1.a.1" aria-label="Copy heading and permalink for Solution 1.a.1">🔗</a></div>
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</div></details></div>
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<div class="autopermalink" data-description="Task 1.a"><a href="#quiz-01-3-1-2" title="Copy heading and permalink for Task 1.a" aria-label="Copy heading and permalink for Task 1.a">🔗</a></div></article><article class="task exercise-like" id="quiz-01-3-1-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
|
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<div class="para" id="quiz-01-3-1-3-1-1">If <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> are any events, then <span class="process-math">\(\Pr(A \mid B) = 1 - \Pr(B \mid A)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-1-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="solutions"><details id="quiz-01-3-1-3-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
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<div class="para" id="quiz-01-3-1-3-2-1">False.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-1-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Task 1.b"><a href="#quiz-01-3-1-3" title="Copy heading and permalink for Task 1.b" aria-label="Copy heading and permalink for Task 1.b">🔗</a></div></article><article class="task exercise-like" id="quiz-01-3-1-4"><h4 class="heading"><span class="codenumber">(c)</span></h4>
|
||||
<div class="para" id="quiz-01-3-1-4-1-1">If <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> are any events, then <span class="process-math">\(\Pr(A\cap B)\Pr(B) = \Pr(A \mid B)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-1-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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||||
<div class="solutions"><details id="quiz-01-3-1-4-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
|
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<div class="para" id="quiz-01-3-1-4-2-1">False.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-1-4-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Solution 1.c.1"><a href="#quiz-01-3-1-4-2" title="Copy heading and permalink for Solution 1.c.1" aria-label="Copy heading and permalink for Solution 1.c.1">🔗</a></div>
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</div></details></div>
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<div class="autopermalink" data-description="Task 1.c"><a href="#quiz-01-3-1-4" title="Copy heading and permalink for Task 1.c" aria-label="Copy heading and permalink for Task 1.c">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 1"><a href="#quiz-01-3-1" title="Copy heading and permalink for Worksheet Exercise 1" aria-label="Copy heading and permalink for Worksheet Exercise 1">🔗</a></div></article><article class="exercise exercise-like" id="quiz-01-3-2"><h3 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<div class="introduction" id="quiz-01-3-2-1"><div class="para" id="quiz-01-3-2-1-1">We find a 4-sided die with faces 0, 1, 3, and 5. An experiment consists of rolling the die two times.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div></div>
|
||||
<article class="task exercise-like" id="quiz-01-3-2-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
|
||||
<div class="para" id="quiz-01-3-2-2-1-1">Write down the sample space <span class="process-math">\(\Omega\)</span> of all possible outcomes for this experiment.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-2-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="solutions"><details id="quiz-01-3-2-2-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
|
||||
<div class="para logical" id="quiz-01-3-2-2-2-1">
|
||||
<div class="displaymath process-math" id="quiz-01-3-2-2-2-1-1">
|
||||
\begin{align*}
|
||||
\Omega = \{ \amp (0, 0), (0, 1), (0, 3), (0, 5), \\
|
||||
\amp (1, 0), (1, 1), (1, 3), (1, 5), \\
|
||||
\amp (3, 0), (3, 1), (3, 3), (3, 5), \\
|
||||
\amp (5, 0), (5, 1), (5, 3), (5, 5)\}
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-2-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
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</div>
|
||||
<div class="autopermalink" data-description="Solution 2.a.1"><a href="#quiz-01-3-2-2-2" title="Copy heading and permalink for Solution 2.a.1" aria-label="Copy heading and permalink for Solution 2.a.1">🔗</a></div>
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||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 2.a"><a href="#quiz-01-3-2-2" title="Copy heading and permalink for Task 2.a" aria-label="Copy heading and permalink for Task 2.a">🔗</a></div></article><article class="task exercise-like" id="quiz-01-3-2-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
|
||||
<div class="para" id="quiz-01-3-2-3-1-1">Let <span class="process-math">\(A\)</span> be the event that the second roll is at least twice the value of the first roll. Let m B be the event that the sum of the rolls is odd. Assume the die is fair. List the outcomes in <span class="process-math">\(A\)</span> and calculate <span class="process-math">\(\Pr(A)\text{.}\)</span> List the outcomes in <span class="process-math">\(B\)</span> and calculate <span class="process-math">\(\Pr(B)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-2-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
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</div>
|
||||
<div class="solutions"><details id="quiz-01-3-2-3-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
|
||||
<div class="para logical" id="quiz-01-3-2-3-2-1">
|
||||
<div class="displaymath process-math" id="quiz-01-3-2-3-2-1-1">
|
||||
\begin{align*}
|
||||
A \amp = \{(0, 0), (0, 1), (0, 3), (0, 5), (1, 3), (1, 5)\} \\
|
||||
B \amp = \{ (0, 1), (0, 3), (0, 5), (1, 0), (3, 0), (5, 0) \}
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">Then <span class="process-math">\(\Pr(A) = \frac{|A|}{|\Omega|} = \frac{6}{16} = \frac{3}{8}\text{,}\)</span> and <span class="process-math">\(\Pr(B) = \frac{|B|}{|\Omega|} = \frac{6}{16} = \frac{3}{8}\text{.}\)</span>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-2-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Solution 2.b.1"><a href="#quiz-01-3-2-3-2" title="Copy heading and permalink for Solution 2.b.1" aria-label="Copy heading and permalink for Solution 2.b.1">🔗</a></div>
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</div></details></div>
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<div class="autopermalink" data-description="Task 2.b"><a href="#quiz-01-3-2-3" title="Copy heading and permalink for Task 2.b" aria-label="Copy heading and permalink for Task 2.b">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 2"><a href="#quiz-01-3-2" title="Copy heading and permalink for Worksheet Exercise 2" aria-label="Copy heading and permalink for Worksheet Exercise 2">🔗</a></div></article></section><section class="onepage" id="quiz-01-4"><article class="exercise exercise-like" id="quiz-01-4-1"><h3 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<div class="para" id="quiz-01-4-1-1-1">A diagnostic test is developed to detect a disease present in 1.3% of the population. For a patient who has the disease, the test will accurately give a positive result 62% of the time. When the patient does not have the disease, the test will accurately give a negative result 99.4% of the time.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-4-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para" id="quiz-01-4-1-1-2">For a patient who receives a positive test, what is the probability they have the disease?<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-4-1-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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||||
<div class="solutions"><details id="quiz-01-4-1-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
|
||||
<div class="para logical" id="quiz-01-4-1-2-1">
|
||||
<div class="para">Let <span class="process-math">\(P\)</span> be the event of receiving a positive test result and <span class="process-math">\(D\)</span> be the event of having the disease. The given information is: <span class="process-math">\(\Pr(D) = 0.013, \Pr(P\mid D) = 0.62\text{,}\)</span> and <span class="process-math">\(\Pr(P^c \mid D^c) = 0.994\text{.}\)</span> Then, using Bayes’ Theorem:</div>
|
||||
<div class="displaymath process-math" id="quiz-01-4-1-2-1-5">
|
||||
\begin{align*}
|
||||
\Pr(D\mid P) = \frac{\Pr(P \mid D)\Pr(D)}{\Pr(P)} \amp = \frac{\Pr(P \mid D)\Pr(D)}{\Pr(P \mid D)\Pr(D) + \Pr(P \mid D^c)\Pr(D^c)} \\
|
||||
\amp = \frac{\Pr(P \mid D)\Pr(D)}{\Pr(P \mid D)\Pr(D) + (1 - \Pr(P^c \mid D^c))(1 - \Pr(D))} \\
|
||||
\amp = \frac{(0.62)(0.013)}{(0.62)(0.013) + (1 - 0.994)(1 - 0.013)} \\
|
||||
\amp \approx \boxed{0.576}
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-4-1-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Solution 3.1"><a href="#quiz-01-4-1-2" title="Copy heading and permalink for Solution 3.1" aria-label="Copy heading and permalink for Solution 3.1">🔗</a></div>
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</div></details></div>
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<div class="autopermalink" data-description="Worksheet Exercise 3"><a href="#quiz-01-4-1" title="Copy heading and permalink for Worksheet Exercise 3" aria-label="Copy heading and permalink for Worksheet Exercise 3">🔗</a></div></article></section><div class="autopermalink" data-description="Worksheet: Quiz 1"><a href="#quiz-01" title="Copy heading and permalink for Worksheet: Quiz 1" aria-label="Copy heading and permalink for Worksheet: Quiz 1">🔗</a></div></section></div>
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<div class="toc-title-box"><a href="notes-01-13.html" class="internal"><span class="title">Tuesday, Jan 13</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Sets" class="internal"><span class="title">Sec 1.1: Sets</span></a></div></li>
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Probability" class="internal"><span class="title">Sec 1.2: Probability</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-13.html#subsec-Sets" class="internal"><span class="title">Sec 1.1: Sets</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-13.html#subsec-Probability" class="internal"><span class="title">Sec 1.2: Probability</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-subsection">
|
||||
<div class="toc-title-box"><a href="notes-week-01.html#notes-week-01-4" class="internal"><span class="title">Thursday 1/15</span></a></div>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-15.html" class="internal"><span class="title">Thursday, Jan 15</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Conditional-Probability" class="internal"><span class="title">Conditional Probability</span></a></div></li>
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Diagnostic-Testing" class="internal"><span class="title">Diagnostic Testing</span></a></div></li>
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Independent-Events" class="internal"><span class="title">Independent Events</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Conditional-Probability" class="internal"><span class="title">Sec 1.3: Conditional Probability</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Diagnostic-Testing" class="internal"><span class="title">Diagnostic Testing</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Independent-Events" class="internal"><span class="title">Sec 1.4: Independent Events</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-20.html#subsec-Discrete-Random-Variables" class="internal"><span class="title">Sec 2.1: Random Variables</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="quizzes.html" class="internal"><span class="title">Quizzes</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-01.html" class="internal"><span class="title">Quiz 1</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter contains-active">
|
||||
<div class="toc-title-box"><a href="recitations.html" class="internal"><span class="title">Recitations</span></a></div>
|
||||
@@ -204,9 +220,9 @@ eBookConfig.enable_chatcodes = false;
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||||
<button class="print-button" title="Print" onClick="window.print()"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Print</span></button>
|
||||
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|
||||
</div>
|
||||
<section class="worksheet" id="recitation-calculus-review" data-margins="0.75in 0.75in 0.75in 0.75in"><h2 class="heading hide-type">
|
||||
<section class="worksheet" id="recitation-calculus-review" data-margins="0.75in 0.75in 0.75in 0.75in"><h1 class="heading hide-type">
|
||||
<span class="type">Worksheet</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Calculus Review</span><div class="print-links"><a href="recitation-calculus-review-printable.html" class="print-link" title="Print preview"><span class="icon material-symbols-outlined" aria-hidden="true"></span></a></div>
|
||||
</h2>
|
||||
</h1>
|
||||
<section class="introduction" id="recitation-calculus-review-2"><div class="para" id="recitation-calculus-review-2-1">In Calculus I, you learned about derivatives, antiderivatives, and definite integrals. All of these will make an appearance in the context of probability density functions, cumulative distribution functions, and probability calculations in continuous sample spaces, so it will be useful to refresh your memory and practice some calculations.<div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></section><section class="onepage" id="recitation-calculus-review-3"><article class="exercise exercise-like" id="recitation-calculus-review-3-1"><h3 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<div class="para logical" id="recitation-calculus-review-3-1-1-1">
|
||||
@@ -255,7 +271,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
</div>
|
||||
<div class="workspace" data-space="2cm"></div>
|
||||
<div class="autopermalink" data-description="Task 4.b"><a href="#recitation-calculus-review-3-4-3" title="Copy heading and permalink for Task 4.b" aria-label="Copy heading and permalink for Task 4.b">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 4"><a href="#recitation-calculus-review-3-4" title="Copy heading and permalink for Worksheet Exercise 4" aria-label="Copy heading and permalink for Worksheet Exercise 4">🔗</a></div></article></section><section class="onepage" id="recitation-calculus-review-4"><article class="theorem theorem-like" id="thm-FTC"><h3 class="heading">
|
||||
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">34</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Fundamental Theorem of Calculus.</span>
|
||||
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">40</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Fundamental Theorem of Calculus.</span>
|
||||
</h3>
|
||||
<div class="para logical" id="thm-FTC-2-1">
|
||||
<div class="para">If <span class="process-math">\(F(x)\)</span> is an antiderivative of <span class="process-math">\(f(x)\)</span>—i.e., if <span class="process-math">\(F'(x) = f(x)\)</span>—then:</div>
|
||||
@@ -266,7 +282,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#thm-FTC-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Theorem 34: Fundamental Theorem of Calculus"><a href="#thm-FTC" title="Copy heading and permalink for Theorem 34: Fundamental Theorem of Calculus" aria-label="Copy heading and permalink for Theorem 34: Fundamental Theorem of Calculus">🔗</a></div></article><div class="para" id="recitation-calculus-review-4-2">If <span class="process-math">\(f(x)\)</span> is a probability density function (don’t worry about what that means for now), then integrals such as <span class="process-math">\(\int_a^b f(x)\ dx\)</span> are used to calculate certain probabilities, as are values of the form <span class="process-math">\(F(x)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<div class="autopermalink" data-description="Theorem 40: Fundamental Theorem of Calculus"><a href="#thm-FTC" title="Copy heading and permalink for Theorem 40: Fundamental Theorem of Calculus" aria-label="Copy heading and permalink for Theorem 40: Fundamental Theorem of Calculus">🔗</a></div></article><div class="para" id="recitation-calculus-review-4-2">If <span class="process-math">\(f(x)\)</span> is a probability density function (don’t worry about what that means for now), then integrals such as <span class="process-math">\(\int_a^b f(x)\ dx\)</span> are used to calculate certain probabilities, as are values of the form <span class="process-math">\(F(x)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="exercise exercise-like" id="recitation-calculus-review-4-3"><h3 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<div class="introduction" id="recitation-calculus-review-4-3-1"><div class="para" id="recitation-calculus-review-4-3-1-1">Practice with the following integrals:<div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
|
||||
@@ -101,15 +101,15 @@ eBookConfig.allow_pairs = false;
|
||||
eBookConfig.enableScratchAC = false;
|
||||
eBookConfig.build_info = "";
|
||||
eBookConfig.python3 = null;
|
||||
eBookConfig.runestone_version = '7.11.9';
|
||||
eBookConfig.runestone_version = '7.11.10';
|
||||
eBookConfig.jobehost = '';
|
||||
eBookConfig.proxyuri_runs = '';
|
||||
eBookConfig.proxyuri_files = '';
|
||||
eBookConfig.enable_chatcodes = false;
|
||||
</script>
|
||||
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|
||||
<script src="_static/prefix-runtime.991c135ccf182bf5.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.1c010553c35c544c.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
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<link rel="stylesheet" type="text/css" href="_static/prefix-runestone.677a75a9120449ff.css">
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<script src="_static/prefix-runtime.3dca4b99808f4053.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.80892ed1280c2803.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
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<link rel="stylesheet" type="text/css" href="_static/prefix-runestone.6ec5960970d8b364.css">
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<script src="_static/pretext/js/lti_iframe_resizer.js"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min.js" integrity="sha512-4xUl/d6D6THrAnXAwGajXkoWaeMNwEKK4iNfq5DotEbLPAfk6FSxSP3ydNxqDgCw1c/0Z1Jg6L8h2j+++9BZmg==" crossorigin="anonymous" referrerpolicy="no-referrer"></script><script src="lunr-pretext-search-index.js" async=""></script><script src="_static/pretext/js/pretext_search.js"></script><script src="_static/pretext/js/lib/knowl.js"></script><!--knowl.js code controls Sage Cells within knowls--><script>sagecellEvalName='Evaluate (Sage)';
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||||
</script>
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||||
</head>
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||||
@@ -140,6 +140,17 @@ eBookConfig.enable_chatcodes = false;
|
||||
\newcommand{\Z}{\mathbb Z}
|
||||
\newcommand{\Q}{\mathbb Q}
|
||||
\newcommand{\R}{\mathbb R}
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||||
|
||||
|
||||
\DeclareMathOperator{\Bin}{Bin}
|
||||
\DeclareMathOperator{\Geom}{Geom}
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||||
\DeclareMathOperator{\Poiss}{Poiss}
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\DeclareMathOperator{\Exp}{Exp}
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||||
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||||
\DeclareMathOperator{\E}{E}
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||||
\DeclareMathOperator{\Var}{Var}
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||||
\DeclareMathOperator{\Cov}{Cov}
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||||
\newcommand{\gt}{>}
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||||
\newcommand{\amp}{&}
|
||||
@@ -163,26 +174,31 @@ eBookConfig.enable_chatcodes = false;
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="ch-notes.html" class="internal"><span class="title">Class Notes</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-week-01.html" class="internal"><span class="title">Week 1</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection">
|
||||
<div class="toc-title-box"><a href="notes-week-01.html#notes-week-01-3" class="internal"><span class="title">Tuesday 1/13</span></a></div>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-13.html" class="internal"><span class="title">Tuesday, Jan 13</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Sets" class="internal"><span class="title">Sec 1.1: Sets</span></a></div></li>
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Probability" class="internal"><span class="title">Sec 1.2: Probability</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-13.html#subsec-Sets" class="internal"><span class="title">Sec 1.1: Sets</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-13.html#subsec-Probability" class="internal"><span class="title">Sec 1.2: Probability</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-subsection">
|
||||
<div class="toc-title-box"><a href="notes-week-01.html#notes-week-01-4" class="internal"><span class="title">Thursday 1/15</span></a></div>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-15.html" class="internal"><span class="title">Thursday, Jan 15</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Conditional-Probability" class="internal"><span class="title">Conditional Probability</span></a></div></li>
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Diagnostic-Testing" class="internal"><span class="title">Diagnostic Testing</span></a></div></li>
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Independent-Events" class="internal"><span class="title">Independent Events</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Conditional-Probability" class="internal"><span class="title">Sec 1.3: Conditional Probability</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Diagnostic-Testing" class="internal"><span class="title">Diagnostic Testing</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Independent-Events" class="internal"><span class="title">Sec 1.4: Independent Events</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-20.html#subsec-Discrete-Random-Variables" class="internal"><span class="title">Sec 2.1: Random Variables</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="quizzes.html" class="internal"><span class="title">Quizzes</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-01.html" class="internal"><span class="title">Quiz 1</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter contains-active">
|
||||
<div class="toc-title-box"><a href="recitations.html" class="internal"><span class="title">Recitations</span></a></div>
|
||||
@@ -240,7 +256,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
</div>
|
||||
<div class="workspace" data-space="2cm"></div>
|
||||
<div class="autopermalink" data-description="Task 4.b"><a href="#recitation-calculus-review-3-4-3" title="Copy heading and permalink for Task 4.b" aria-label="Copy heading and permalink for Task 4.b">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 4"><a href="#recitation-calculus-review-3-4" title="Copy heading and permalink for Worksheet Exercise 4" aria-label="Copy heading and permalink for Worksheet Exercise 4">🔗</a></div></article></section><section class="onepage" id="recitation-calculus-review-4"><article class="theorem theorem-like" id="thm-FTC"><h3 class="heading">
|
||||
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">34</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Fundamental Theorem of Calculus.</span>
|
||||
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">40</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Fundamental Theorem of Calculus.</span>
|
||||
</h3>
|
||||
<div class="para logical" id="thm-FTC-2-1">
|
||||
<div class="para">If <span class="process-math">\(F(x)\)</span> is an antiderivative of <span class="process-math">\(f(x)\)</span>—i.e., if <span class="process-math">\(F'(x) = f(x)\)</span>—then:</div>
|
||||
@@ -251,7 +267,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#thm-FTC-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Theorem 34: Fundamental Theorem of Calculus"><a href="#thm-FTC" title="Copy heading and permalink for Theorem 34: Fundamental Theorem of Calculus" aria-label="Copy heading and permalink for Theorem 34: Fundamental Theorem of Calculus">🔗</a></div></article><div class="para" id="recitation-calculus-review-4-2">If <span class="process-math">\(f(x)\)</span> is a probability density function (don’t worry about what that means for now), then integrals such as <span class="process-math">\(\int_a^b f(x)\ dx\)</span> are used to calculate certain probabilities, as are values of the form <span class="process-math">\(F(x)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<div class="autopermalink" data-description="Theorem 40: Fundamental Theorem of Calculus"><a href="#thm-FTC" title="Copy heading and permalink for Theorem 40: Fundamental Theorem of Calculus" aria-label="Copy heading and permalink for Theorem 40: Fundamental Theorem of Calculus">🔗</a></div></article><div class="para" id="recitation-calculus-review-4-2">If <span class="process-math">\(f(x)\)</span> is a probability density function (don’t worry about what that means for now), then integrals such as <span class="process-math">\(\int_a^b f(x)\ dx\)</span> are used to calculate certain probabilities, as are values of the form <span class="process-math">\(F(x)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="exercise exercise-like" id="recitation-calculus-review-4-3"><h3 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<div class="introduction" id="recitation-calculus-review-4-3-1"><div class="para" id="recitation-calculus-review-4-3-1-1">Practice with the following integrals:<div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
|
||||
+33
-17
@@ -101,15 +101,15 @@ eBookConfig.allow_pairs = false;
|
||||
eBookConfig.enableScratchAC = false;
|
||||
eBookConfig.build_info = "";
|
||||
eBookConfig.python3 = null;
|
||||
eBookConfig.runestone_version = '7.11.9';
|
||||
eBookConfig.runestone_version = '7.11.10';
|
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eBookConfig.proxyuri_runs = '';
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eBookConfig.proxyuri_files = '';
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</head>
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<button id="light-dark-button" class="light-dark-button button" title="Dark Mode"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Dark Mode</span></button></span><span class="treebuttons"><a class="previous-button button" href="quiz-01.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Prev</span></a><a class="up-button button" href="root-1-2.html" title="Up"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Up</span></a><a class="next-button button" href="recitation-calculus-review.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></a></span>
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</div></nav><div id="latex-macros" class="hidden-content process-math" style="display:none"><span class="process-math">\(\newcommand{\N}{\mathbb N}
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<div class="toc-title-box"><a href="ch-notes.html" class="internal"><span class="title">Class Notes</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-section">
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<li class="toc-item toc-subsection">
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||||
<div class="toc-title-box"><a href="notes-week-01.html#notes-week-01-3" class="internal"><span class="title">Tuesday 1/13</span></a></div>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-13.html" class="internal"><span class="title">Tuesday, Jan 13</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Sets" class="internal"><span class="title">Sec 1.1: Sets</span></a></div></li>
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Probability" class="internal"><span class="title">Sec 1.2: Probability</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-13.html#subsec-Sets" class="internal"><span class="title">Sec 1.1: Sets</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-13.html#subsec-Probability" class="internal"><span class="title">Sec 1.2: Probability</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-subsection">
|
||||
<div class="toc-title-box"><a href="notes-week-01.html#notes-week-01-4" class="internal"><span class="title">Thursday 1/15</span></a></div>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-15.html" class="internal"><span class="title">Thursday, Jan 15</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Conditional-Probability" class="internal"><span class="title">Conditional Probability</span></a></div></li>
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Diagnostic-Testing" class="internal"><span class="title">Diagnostic Testing</span></a></div></li>
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Independent-Events" class="internal"><span class="title">Independent Events</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Conditional-Probability" class="internal"><span class="title">Sec 1.3: Conditional Probability</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Diagnostic-Testing" class="internal"><span class="title">Diagnostic Testing</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Independent-Events" class="internal"><span class="title">Sec 1.4: Independent Events</span></a></div></li>
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||||
</ul>
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</li>
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<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-20.html#subsec-Discrete-Random-Variables" class="internal"><span class="title">Sec 2.1: Random Variables</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li></ul>
|
||||
</li>
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<li class="toc-item toc-chapter">
|
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<div class="toc-title-box"><a href="quizzes.html" class="internal"><span class="title">Quizzes</span></a></div>
|
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<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-01.html" class="internal"><span class="title">Quiz 1</span></a></div></li></ul>
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</li>
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<div class="toc-title-box"><a href="recitations.html" class="internal"><span class="title">Recitations</span></a></div>
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@@ -194,7 +210,7 @@ eBookConfig.enable_chatcodes = false;
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<nav class="summary-links"><ul><li><a href="recitation-calculus-review.html" class="internal"><span class="title">Calculus Review</span></a></li></ul></nav><div class="autopermalink" data-description="Chapter: Recitations"><a href="#recitations" title="Copy heading and permalink for Chapter: Recitations" aria-label="Copy heading and permalink for Chapter: Recitations">🔗</a></div></section></div>
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<a class="previous-button button" href="notes-week-01.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Prev</span></a><a class="top-button button" href="#" title="Top"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Top</span></a><a class="next-button button" href="recitation-calculus-review.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></a>
|
||||
<a class="previous-button button" href="quiz-01.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Prev</span></a><a class="top-button button" href="#" title="Top"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Top</span></a><a class="next-button button" href="recitation-calculus-review.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></a>
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||||
</div></main>
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</div>
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<div id="ptx-page-footer" class="ptx-page-footer">
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+32
-15
@@ -101,15 +101,15 @@ eBookConfig.allow_pairs = false;
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eBookConfig.enableScratchAC = false;
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eBookConfig.build_info = "";
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eBookConfig.python3 = null;
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eBookConfig.runestone_version = '7.11.9';
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eBookConfig.runestone_version = '7.11.10';
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eBookConfig.jobehost = '';
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eBookConfig.proxyuri_runs = '';
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eBookConfig.proxyuri_files = '';
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eBookConfig.enable_chatcodes = false;
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<script src="_static/prefix-runtime.991c135ccf182bf5.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.1c010553c35c544c.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
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<script src="_static/prefix-runtime.3dca4b99808f4053.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.80892ed1280c2803.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
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<link rel="stylesheet" type="text/css" href="_static/prefix-runestone.6ec5960970d8b364.css">
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<script src="_static/pretext/js/lti_iframe_resizer.js"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min.js" integrity="sha512-4xUl/d6D6THrAnXAwGajXkoWaeMNwEKK4iNfq5DotEbLPAfk6FSxSP3ydNxqDgCw1c/0Z1Jg6L8h2j+++9BZmg==" crossorigin="anonymous" referrerpolicy="no-referrer"></script><script src="lunr-pretext-search-index.js" async=""></script><script src="_static/pretext/js/pretext_search.js"></script><script src="_static/pretext/js/lib/knowl.js"></script><!--knowl.js code controls Sage Cells within knowls--><script>sagecellEvalName='Evaluate (Sage)';
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</head>
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||||
@@ -140,6 +140,17 @@ eBookConfig.enable_chatcodes = false;
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||||
\newcommand{\Z}{\mathbb Z}
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\DeclareMathOperator{\Bin}{Bin}
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\DeclareMathOperator{\Geom}{Geom}
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\DeclareMathOperator{\Poiss}{Poiss}
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\DeclareMathOperator{\Exp}{Exp}
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\DeclareMathOperator{\E}{E}
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\DeclareMathOperator{\Var}{Var}
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\DeclareMathOperator{\Cov}{Cov}
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\newcommand{\lt}{<}
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\newcommand{\gt}{>}
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\newcommand{\amp}{&}
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||||
@@ -163,26 +174,31 @@ eBookConfig.enable_chatcodes = false;
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||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="ch-notes.html" class="internal"><span class="title">Class Notes</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-week-01.html" class="internal"><span class="title">Week 1</span></a></div>
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||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection">
|
||||
<div class="toc-title-box"><a href="notes-week-01.html#notes-week-01-3" class="internal"><span class="title">Tuesday 1/13</span></a></div>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-13.html" class="internal"><span class="title">Tuesday, Jan 13</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Sets" class="internal"><span class="title">Sec 1.1: Sets</span></a></div></li>
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Probability" class="internal"><span class="title">Sec 1.2: Probability</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-13.html#subsec-Sets" class="internal"><span class="title">Sec 1.1: Sets</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-13.html#subsec-Probability" class="internal"><span class="title">Sec 1.2: Probability</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-subsection">
|
||||
<div class="toc-title-box"><a href="notes-week-01.html#notes-week-01-4" class="internal"><span class="title">Thursday 1/15</span></a></div>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-15.html" class="internal"><span class="title">Thursday, Jan 15</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Conditional-Probability" class="internal"><span class="title">Conditional Probability</span></a></div></li>
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Diagnostic-Testing" class="internal"><span class="title">Diagnostic Testing</span></a></div></li>
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Independent-Events" class="internal"><span class="title">Independent Events</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Conditional-Probability" class="internal"><span class="title">Sec 1.3: Conditional Probability</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Diagnostic-Testing" class="internal"><span class="title">Diagnostic Testing</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Independent-Events" class="internal"><span class="title">Sec 1.4: Independent Events</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-20.html#subsec-Discrete-Random-Variables" class="internal"><span class="title">Sec 2.1: Random Variables</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="quizzes.html" class="internal"><span class="title">Quizzes</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-01.html" class="internal"><span class="title">Quiz 1</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="recitations.html" class="internal"><span class="title">Recitations</span></a></div>
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<li><a href="course-documents.html" class="internal"><span class="title">Course Documents</span></a></li>
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<li><a href="ch-notes.html" class="internal"><span class="title">Class Notes</span></a></li>
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<li><a href="recitations.html" class="internal"><span class="title">Recitations</span></a></li>
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<div id="ptx-content-footer" class="ptx-content-footer">
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+31
-15
@@ -101,15 +101,15 @@ eBookConfig.allow_pairs = false;
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|
||||
</script>
|
||||
</head>
|
||||
@@ -140,6 +140,17 @@ eBookConfig.enable_chatcodes = false;
|
||||
\newcommand{\Z}{\mathbb Z}
|
||||
\newcommand{\Q}{\mathbb Q}
|
||||
\newcommand{\R}{\mathbb R}
|
||||
|
||||
|
||||
\DeclareMathOperator{\Bin}{Bin}
|
||||
\DeclareMathOperator{\Geom}{Geom}
|
||||
\DeclareMathOperator{\Poiss}{Poiss}
|
||||
\DeclareMathOperator{\Exp}{Exp}
|
||||
|
||||
|
||||
\DeclareMathOperator{\E}{E}
|
||||
\DeclareMathOperator{\Var}{Var}
|
||||
\DeclareMathOperator{\Cov}{Cov}
|
||||
\newcommand{\lt}{<}
|
||||
\newcommand{\gt}{>}
|
||||
\newcommand{\amp}{&}
|
||||
@@ -163,26 +174,31 @@ eBookConfig.enable_chatcodes = false;
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="ch-notes.html" class="internal"><span class="title">Class Notes</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-week-01.html" class="internal"><span class="title">Week 1</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection">
|
||||
<div class="toc-title-box"><a href="notes-week-01.html#notes-week-01-3" class="internal"><span class="title">Tuesday 1/13</span></a></div>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-13.html" class="internal"><span class="title">Tuesday, Jan 13</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Sets" class="internal"><span class="title">Sec 1.1: Sets</span></a></div></li>
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Probability" class="internal"><span class="title">Sec 1.2: Probability</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-13.html#subsec-Sets" class="internal"><span class="title">Sec 1.1: Sets</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-13.html#subsec-Probability" class="internal"><span class="title">Sec 1.2: Probability</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-subsection">
|
||||
<div class="toc-title-box"><a href="notes-week-01.html#notes-week-01-4" class="internal"><span class="title">Thursday 1/15</span></a></div>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-15.html" class="internal"><span class="title">Thursday, Jan 15</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Conditional-Probability" class="internal"><span class="title">Conditional Probability</span></a></div></li>
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Diagnostic-Testing" class="internal"><span class="title">Diagnostic Testing</span></a></div></li>
|
||||
<li class="toc-item toc-subsubsection"><div class="toc-title-box"><a href="notes-week-01.html#subsubsec-Independent-Events" class="internal"><span class="title">Independent Events</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Conditional-Probability" class="internal"><span class="title">Sec 1.3: Conditional Probability</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Diagnostic-Testing" class="internal"><span class="title">Diagnostic Testing</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Independent-Events" class="internal"><span class="title">Sec 1.4: Independent Events</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-20.html#subsec-Discrete-Random-Variables" class="internal"><span class="title">Sec 2.1: Random Variables</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="quizzes.html" class="internal"><span class="title">Quizzes</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-01.html" class="internal"><span class="title">Quiz 1</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="recitations.html" class="internal"><span class="title">Recitations</span></a></div>
|
||||
|
||||
Reference in New Issue
Block a user