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@@ -165,13 +165,23 @@ eBookConfig.enable_chatcodes = false;
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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 contains-active"><li class="toc-item toc-section contains-active">
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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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<ul class="structural toc-item-list">
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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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</ul>
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</li></ul>
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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-01-4" class="internal"><span class="title">Thursday 1/15</span></a></div>
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<ul class="structural toc-item-list">
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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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</ul>
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@@ -489,7 +499,205 @@ eBookConfig.enable_chatcodes = false;
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<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>
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</div>
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<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><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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<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">
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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">Thursday 1/15</span>
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</h3>
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<section class="subsubsection" id="subsubsec-Conditional-Probability"><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">Conditional Probability</span>
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</h4>
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<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>
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</div>
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<article class="example example-like" id="subsubsec-Conditional-Probability-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">22</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-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>
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</div>
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<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">
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<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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</h5>
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<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>
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<div class="displaymath process-math" id="def-conditional-probability-1-1-4">
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\begin{gather*}
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\Pr(A \mid B) = \frac{\Pr(A\cap B)}{\Pr(B)}
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\end{gather*}
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</div>
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<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>
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</div> <figure class="figure figure-like" id="fig-conditional-probability"><div class="image-box" style="width: 50%; margin-left: 25%; margin-right: 25%;">
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<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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</div>
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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"><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">25</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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</h5>
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<div class="para logical" id="example-rolls-conditional-1-1">
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<div class="para">Continuing from the previous example, <span class="process-math">\(|\Omega| = 36\text{.}\)</span>
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</div>
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<div class="displaymath process-math" id="example-rolls-conditional-1-1-2">
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\begin{align*}
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A \amp = \{(4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)\} \\
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B \amp = \{(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\} \\
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A \cap B \amp = \{(6, 4), (6, 5), (6, 6)\}
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\end{align*}
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</div>
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<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>
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<div class="displaymath process-math" id="example-rolls-conditional-1-1-4">
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\begin{gather*}
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\Pr(A \mid B) = \frac{\Pr(A\cap B)}{\Pr(B)} = \frac{3/36}{6/36} = \frac{3}{6} = \frac{1}{2}.
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\end{gather*}
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</div>
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<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>
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</div>
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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>
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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><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">
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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">Diagnostic Testing</span>
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</h4>
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<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>
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</div>
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<article class="definition definition-like" id="def-sensitivity-specificity"><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">26</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="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>
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</div>
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<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>
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</div>
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<article class="example example-like" id="subsubsec-Diagnostic-Testing-5"><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">27</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-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>
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</div> <div class="para logical" id="subsubsec-Diagnostic-Testing-5-1-2">
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<div class="displaymath process-math" id="subsubsec-Diagnostic-Testing-5-1-2-1">
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\begin{align*}
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\text{A) } 9/10 \amp \amp \text{B) } 8/10 \amp \amp \text{C) } 1/10 \amp \amp \text{D) } 1/100
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\end{align*}
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</div>
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<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>
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</div>
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<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">
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<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>
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</div>
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<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>
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</div></details></div>
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<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">
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<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>
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</h5>
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<div class="para logical" id="thm-Bayes-v1-2-1">
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<div class="para">For events <span class="process-math">\(A, B\)</span> with nonzero probability:</div>
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<div class="displaymath process-math" id="thm-Bayes-v1-2-1-2">
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\begin{gather*}
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\Pr(B \mid A) = \frac{\Pr(A \mid B)\Pr(B)}{\Pr(A)}
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\end{gather*}
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</div>
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<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>
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||||
</div>
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||||
<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">
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<div class="para">For example, if a patient sees a positive diagnostic test result, they might try to calculate:</div>
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<div class="displaymath process-math" id="subsubsec-Diagnostic-Testing-7-1">
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\begin{gather*}
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\Pr(D \mid P) = \frac{\Pr(P \mid D)\Pr(D)}{\Pr(P)}
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\end{gather*}
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</div>
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<div class="para">
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<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>
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</div>
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<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>
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</div>
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<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>
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||||
</div>
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||||
<figure class="figure figure-like" id="fig-P-breakdown"><div class="image-box" style="width: 50%; margin-left: 25%; margin-right: 25%;">
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<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>
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||||
</div>
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||||
<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">
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<div class="para">Observation 2:</div>
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<div class="displaymath process-math" id="subsubsec-Diagnostic-Testing-10-1">
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\begin{align*}
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\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) \\
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\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)
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\end{align*}
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</div>
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<div class="para">So:</div>
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<div class="displaymath process-math" id="subsubsec-Diagnostic-Testing-10-2">
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\begin{gather*}
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\Pr(P) = \Pr(P\mid D)\Pr(D) + \Pr(P\mid D^c)\Pr(D^c)
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\end{gather*}
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</div>
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<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>
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||||
</div>
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||||
<article class="theorem theorem-like" id="thm-Bayes-v2"><h5 class="heading">
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<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>
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</h5>
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||||
<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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