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2026-01-11 13:24:27 -05:00
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@@ -9,7 +9,7 @@
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@@ -101,15 +101,15 @@ eBookConfig.allow_pairs = false;
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eBookConfig.build_info = "";
eBookConfig.python3 = null;
eBookConfig.runestone_version = '7.11.5';
eBookConfig.runestone_version = '7.11.8';
eBookConfig.jobehost = '';
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eBookConfig.proxyuri_files = '';
eBookConfig.enable_chatcodes = false;
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@@ -129,7 +129,13 @@ eBookConfig.enable_chatcodes = false;
<ol id="searchresults" class="searchresults"></ol>
</div>
</div>
<span class="nav-other-controls"><button id="light-dark-button" class="light-dark-button button" title="Dark Mode"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe51c;</span><span class="name">Dark Mode</span></button></span><span class="treebuttons"><a class="previous-button button" href="sec-Set-Theory.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe5cb;</span><span class="name">Prev</span></a><a class="up-button button" href="ch-Probability.html" title="Up"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe5ce;</span><span class="name">Up</span></a><a class="next-button button" href="sec-Conditional-Probability.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true">&#xe5cc;</span></a></span>
<span class="nav-other-controls"><button id="embed-button" class="embed-button button" title="Embed this page"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe86f;</span><span class="name">Embed</span></button><div class="embed-popup hidden" id="embed-popup">
<p>Copy the code below to embed this page in your own website or LMS page.</p>
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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">&#xe51c;</span><span class="name">Dark Mode</span></button></span><span class="treebuttons"><a class="previous-button button" href="sec-Set-Theory.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe5cb;</span><span class="name">Prev</span></a><a class="up-button button" href="ch-Probability.html" title="Up"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe5ce;</span><span class="name">Up</span></a><a class="next-button button" href="sec-Conditional-Probability.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true">&#xe5cc;</span></a></span>
</div></nav><div id="latex-macros" class="hidden-content process-math" style="display:none"><span class="process-math">\(\newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb Z}
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@@ -144,6 +150,7 @@ eBookConfig.enable_chatcodes = false;
\DeclareMathOperator{\E}{E}
\DeclareMathOperator{\Var}{Var}
\DeclareMathOperator{\Cov}{Cov}
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@@ -151,8 +158,12 @@ eBookConfig.enable_chatcodes = false;
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\)</span></div>
<div class="ptx-page">
<div id="ptx-sidebar" class="ptx-sidebar"><nav id="ptx-toc" class="ptx-toc depth2 focused" data-preexpanded-levels="0" data-max-levels="2"><ul class="structural toc-item-list contains-active">
<div id="ptx-sidebar" class="ptx-sidebar"><nav id="ptx-toc" class="ptx-toc depth2"><ul class="structural toc-item-list contains-active">
<li class="toc-item toc-frontmatter"><div class="toc-title-box"><a href="frontmatter.html" class="internal"><span class="title">Front Matter</span></a></div></li>
<li class="toc-item toc-chapter">
<div class="toc-title-box"><a href="ch-Introduction.html" class="internal"><span class="codenumber">0</span> <span class="title">Introduction</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Introduction.html" class="internal"><span class="codenumber">0.1</span> <span class="title">Introduction</span></a></div></li></ul>
</li>
<li class="toc-item toc-chapter contains-active">
<div class="toc-title-box"><a href="ch-Probability.html" class="internal"><span class="codenumber">1</span> <span class="title">Probability Theory</span></a></div>
<ul class="structural toc-item-list contains-active">
@@ -201,6 +212,10 @@ eBookConfig.enable_chatcodes = false;
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#exercises-Continuous-RVs" class="internal"><span class="codenumber">2.2.5</span> <span class="title">Exercises</span></a></div></li>
</ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-Joint-Distributions.html" class="internal"><span class="codenumber">2.3</span> <span class="title">Joint Distributions</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Joint-Distributions.html#exercises-Joint-Distributions" class="internal"><span class="codenumber">2.3</span> <span class="title">Exercises</span></a></div></li></ul>
</li>
</ul>
</li>
<li class="toc-item toc-chapter">
@@ -228,24 +243,48 @@ eBookConfig.enable_chatcodes = false;
<li class="toc-item toc-chapter">
<div class="toc-title-box"><a href="ch-Confidence-Intervals.html" class="internal"><span class="codenumber">4</span> <span class="title">Confidence Intervals</span></a></div>
<ul class="structural toc-item-list">
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-CLT.html" class="internal"><span class="codenumber">4.1</span> <span class="title">Central Limit Theorem</span></a></div></li>
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Confidence-Intervals.html" class="internal"><span class="codenumber">4.2</span> <span class="title">Confidence Intervals</span></a></div></li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-CLT.html" class="internal"><span class="codenumber">4.1</span> <span class="title">Central Limit Theorem</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-CLT.html#exercises-CLT" class="internal"><span class="codenumber">4.1</span> <span class="title">Exercises</span></a></div></li></ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-Confidence-Intervals.html" class="internal"><span class="codenumber">4.2</span> <span class="title">Confidence Intervals</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Confidence-Intervals.html#exercises-Confidence-Intervals" class="internal"><span class="codenumber">4.2</span> <span class="title">Exercises</span></a></div></li></ul>
</li>
</ul>
</li>
<li class="toc-item toc-chapter">
<div class="toc-title-box"><a href="ch-Hypothesis-Testing.html" class="internal"><span class="codenumber">5</span> <span class="title">Hypothesis Testing</span></a></div>
<ul class="structural toc-item-list">
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-One-Sample-Tests.html" class="internal"><span class="codenumber">5.1</span> <span class="title">One Sample Tests</span></a></div></li>
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Two-Sample-Tests.html" class="internal"><span class="codenumber">5.2</span> <span class="title">Two Sample Tests</span></a></div></li>
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Power.html" class="internal"><span class="codenumber">5.3</span> <span class="title">Power of a Test</span></a></div></li>
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Chi-Squared.html" class="internal"><span class="codenumber">5.4</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Test</span></a></div></li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-One-Sample-Tests.html" class="internal"><span class="codenumber">5.1</span> <span class="title">One Sample Tests</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-One-Sample-Tests.html#exercises-One-Sample-Tests" class="internal"><span class="codenumber">5.1</span> <span class="title">Exercises</span></a></div></li></ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-Two-Sample-Tests.html" class="internal"><span class="codenumber">5.2</span> <span class="title">Two Sample Tests</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Two-Sample-Tests.html#exercises-Two-Sample-Tests" class="internal"><span class="codenumber">5.2</span> <span class="title">Exercises</span></a></div></li></ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-Power.html" class="internal"><span class="codenumber">5.3</span> <span class="title">Power of a Test</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Power.html#exercises-Power" class="internal"><span class="codenumber">5.3</span> <span class="title">Exercises</span></a></div></li></ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-Chi-Squared.html" class="internal"><span class="codenumber">5.4</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Test</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Chi-Squared.html#exercises-Chi-Squared" class="internal"><span class="codenumber">5.4</span> <span class="title">Exercises</span></a></div></li></ul>
</li>
</ul>
</li>
<li class="toc-item toc-chapter">
<div class="toc-title-box"><a href="ch-Linear-Regression.html" class="internal"><span class="codenumber">6</span> <span class="title">Linear Regression</span></a></div>
<ul class="structural toc-item-list">
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Correlation.html" class="internal"><span class="codenumber">6.1</span> <span class="title">Correlation</span></a></div></li>
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Linear-Regression.html" class="internal"><span class="codenumber">6.2</span> <span class="title">Linear Regression</span></a></div></li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-Correlation.html" class="internal"><span class="codenumber">6.1</span> <span class="title">Correlation</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Correlation.html#exercises-Correlation" class="internal"><span class="codenumber">6.1</span> <span class="title">Exercises</span></a></div></li></ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-Linear-Regression.html" class="internal"><span class="codenumber">6.2</span> <span class="title">Linear Regression</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Linear-Regression.html#exercises-Linear-Regression" class="internal"><span class="codenumber">6.2</span> <span class="title">Exercises</span></a></div></li></ul>
</li>
</ul>
</li>
<li class="toc-item toc-backmatter"><div class="toc-title-box"><a href="backmatter.html" class="internal"><span class="title">Backmatter</span></a></div></li>
@@ -253,6 +292,8 @@ eBookConfig.enable_chatcodes = false;
<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="sec-Probability"><h2 class="heading hide-type">
<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.2</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Definition of Probability</span>
</h2>
<div class="para" id="sec-Probability-2">Now that we have the language to refer to outcomes and events of an experiment, we want to start quantifying how likely those outcomes/events are to occur.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Probability-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-probability-distribution"><h3 class="heading">
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.2.1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h3>
@@ -279,13 +320,13 @@ eBookConfig.enable_chatcodes = false;
</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 1.2.1"><a href="#def-probability-distribution" title="Copy heading and permalink for Definition 1.2.1" aria-label="Copy heading and permalink for Definition 1.2.1">🔗</a></div></article><div class="para" id="sec-Probability-3">For small probability spaces (i.e., with finitely many outcomes in the sample space), we’ll usually assign probabilities to each individual outcome, and perhaps list them in a table. Then, to find the probability of any event, simply add together the probabilities of each outcome in that event.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Probability-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="autopermalink" data-description="Definition 1.2.1"><a href="#def-probability-distribution" title="Copy heading and permalink for Definition 1.2.1" aria-label="Copy heading and permalink for Definition 1.2.1">🔗</a></div></article><div class="para" id="sec-Probability-4">For small probability spaces (i.e., with finitely many outcomes in the sample space), we’ll usually assign probabilities to each individual outcome, and perhaps list them in a table. Then, to find the probability of any event, simply add together the probabilities of each outcome in that event.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Probability-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="sec-Probability-4"><h3 class="heading">
<article class="example example-like" id="sec-Probability-5"><h3 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.2.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h3>
<div class="para" id="sec-Probability-4-1-1">An experiment consists of rolling a standard 6-sided die. The sample space is <span class="process-math">\(\Omega = \{1, 2, 3, 4, 5, 6\}\text{.}\)</span> The probability distribution (assuming a fair die) is shown below.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Probability-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="sec-Probability-4-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.2.3<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 1.2.3: Distribution for a fair die"><a href="#sec-Probability-4-1-2" title="Copy heading and permalink for Table 1.2.3: Distribution for a fair die" aria-label="Copy heading and permalink for Table 1.2.3: Distribution for a fair die">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<div class="para" id="sec-Probability-5-1-1">An experiment consists of rolling a standard 6-sided die. The sample space is <span class="process-math">\(\Omega = \{1, 2, 3, 4, 5, 6\}\text{.}\)</span> The probability distribution (assuming a fair die) is shown below.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Probability-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="sec-Probability-5-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.2.3<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 1.2.3: Distribution for a fair die"><a href="#sec-Probability-5-1-2" title="Copy heading and permalink for Table 1.2.3: Distribution for a fair die" aria-label="Copy heading and permalink for Table 1.2.3: 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>
@@ -314,26 +355,35 @@ eBookConfig.enable_chatcodes = false;
<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="sec-Probability-4-1-3">
</table></div></figure> <div class="para logical" id="sec-Probability-5-1-3">
<div class="para">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. The probability of <span class="process-math">\(A\)</span> is:</div>
<div class="displaymath process-math" id="sec-Probability-4-1-3-3">
<div class="displaymath process-math" id="sec-Probability-5-1-3-3">
\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="autopermalink" data-description="Paragraph"><a href="#sec-Probability-4-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="autopermalink" data-description="Paragraph"><a href="#sec-Probability-5-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 1.2.2"><a href="#sec-Probability-4" title="Copy heading and permalink for Example 1.2.2" aria-label="Copy heading and permalink for Example 1.2.2">🔗</a></div></article><article class="definition definition-like" id="def-discrete-uniform-distribution"><h3 class="heading">
<div class="autopermalink" data-description="Example 1.2.2"><a href="#sec-Probability-5" title="Copy heading and permalink for Example 1.2.2" aria-label="Copy heading and permalink for Example 1.2.2">🔗</a></div></article><article class="definition definition-like" id="def-discrete-uniform-distribution"><h3 class="heading">
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.2.4</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h3>
<div class="para" id="def-discrete-uniform-distribution-1-1">Let <span class="process-math">\(\Omega = \{x_1, x_2, \dotsc, x_n\}\)</span> be a sample space and <span class="process-math">\(A\subset \Omega\)</span> an event. We refer to the distribution in which <span class="process-math">\(\Pr(x_i) = \frac{1}{n}\)</span> for all <span class="process-math">\(i\)</span> as the <dfn class="terminology">uniform distribution</dfn>. In this case, it follows that <span class="process-math">\(\Pr(A) = \frac{|A|}{|\Omega|}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-discrete-uniform-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 1.2.4"><a href="#def-discrete-uniform-distribution" title="Copy heading and permalink for Definition 1.2.4" aria-label="Copy heading and permalink for Definition 1.2.4">🔗</a></div></article><article class="example example-like" id="sec-Probability-6"><h3 class="heading">
<div class="autopermalink" data-description="Definition 1.2.4"><a href="#def-discrete-uniform-distribution" title="Copy heading and permalink for Definition 1.2.4" aria-label="Copy heading and permalink for Definition 1.2.4">🔗</a></div></article><div class="para" id="sec-Probability-7">When we talk about a fair coin flip or a fair die roll, the word "fair" is indicating a uniform distribution. However, don’t make the mistake of assuming that all distributions are uniform by default.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Probability-7" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<article class="example example-like" id="sec-Probability-8"><h3 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.2.5</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h3>
<div class="para" id="sec-Probability-6-1-1">A person picks a random number from 1 to 10. What is the probability that they picked 3?<div class="autopermalink" data-description="Paragraph"><a href="#sec-Probability-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="para" id="sec-Probability-8-1-1">A person picks a random number from 1 to 10. What is the probability that they picked 3?<div class="autopermalink" data-description="Paragraph"><a href="#sec-Probability-8-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 1.2.5"><a href="#sec-Probability-6" title="Copy heading and permalink for Example 1.2.5" aria-label="Copy heading and permalink for Example 1.2.5">🔗</a></div></article><section class="exercises" id="exercises-Probability"><h3 class="heading hide-type">
<div class="solutions"><details id="sec-Probability-8-2" 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="sec-Probability-8-2-1">Without assuming the distribution is fair (i.e., that each value <span class="process-math">\(1, 2, \dotsc, 10\)</span>) has probability <span class="process-math">\(1/10\)</span> of occurring), we don’t have enough information to answer this question.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Probability-8-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="para" id="sec-Probability-8-2-2">In fact, the situation is even more vague than that: the sample space itself is unclear. Are we only allowed to pick integer values? What about fractions like <span class="process-math">\(7/2\text{?}\)</span> What about irrational numbers like <span class="process-math">\(\pi\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#sec-Probability-8-2-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 1.2.5.1"><a href="#sec-Probability-8-2" title="Copy heading and permalink for Answer 1.2.5.1" aria-label="Copy heading and permalink for Answer 1.2.5.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Example 1.2.5"><a href="#sec-Probability-8" title="Copy heading and permalink for Example 1.2.5" aria-label="Copy heading and permalink for Example 1.2.5">🔗</a></div></article><section class="exercises" id="exercises-Probability"><h3 class="heading hide-type">
<span class="type">Exercises</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">Exercises</span>
</h3>
<article class="exercise exercise-like" id="exercises-Probability-1"><h4 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
@@ -379,43 +429,214 @@ eBookConfig.enable_chatcodes = false;
<td class="c m b0 r0 l0 t0 lines">9</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
</table></div></figure> <div class="autopermalink" data-description="Exercise 1.2.1"><a href="#exercises-Probability-1" title="Copy heading and permalink for Exercise 1.2.1" aria-label="Copy heading and permalink for Exercise 1.2.1">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Probability-2"><h4 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
</table></div></figure><div class="solutions"><details id="exercises-Probability-1-2" 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="exercises-Probability-1-2-1">
<span class="process-math">\(\Pr(A) = 0.45, \Pr(B) = 0.4, \Pr(A \cup B) = 0.6, \Pr(A \cap B) = 0.25.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-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="Answer 1.2.1.1"><a href="#exercises-Probability-1-2" title="Copy heading and permalink for Answer 1.2.1.1" aria-label="Copy heading and permalink for Answer 1.2.1.1">🔗</a></div>
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<div class="autopermalink" data-description="Exercise 1.2.1"><a href="#exercises-Probability-1" title="Copy heading and permalink for Exercise 1.2.1" aria-label="Copy heading and permalink for Exercise 1.2.1">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Probability-2"><h4 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="introduction" id="exercises-Probability-2-1"><div class="para" id="exercises-Probability-2-1-1">Suppose we flip a coin two times. Answer the questions below. What about three flips? What about four flips?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></div>
<article class="task exercise-like" id="exercises-Probability-2-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
<div class="para" id="exercises-Probability-2-2-1-1">Write all outcomes in the sample space <span class="process-math">\(\Omega\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-2-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="exercises-Probability-2-2-2" 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="exercises-Probability-2-2-2-1">
<span class="process-math">\(\Omega = \{HH, HT, TH, TT\}.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-2-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 1.2.2.a.1"><a href="#exercises-Probability-2-2-2" title="Copy heading and permalink for Answer 1.2.2.a.1" aria-label="Copy heading and permalink for Answer 1.2.2.a.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 1.2.2.a"><a href="#exercises-Probability-2-2" title="Copy heading and permalink for Task 1.2.2.a" aria-label="Copy heading and permalink for Task 1.2.2.a">🔗</a></div></article><article class="task exercise-like" id="exercises-Probability-2-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
<div class="para" id="exercises-Probability-2-3-1-1">Make a probability distribution table for <span class="process-math">\(\Omega\)</span> assuming the coin is fair.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-2-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="exercises-Probability-2-3-2" 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">
<figure class="table table-like" id="exercises-Probability-2-3-2-1"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.2.7<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Probability Distribution for Two Fair Coin Flips<div class="autopermalink" data-description="Table 1.2.7: Probability Distribution for Two Fair Coin Flips"><a href="#exercises-Probability-2-3-2-1" title="Copy heading and permalink for Table 1.2.7: Probability Distribution for Two Fair Coin Flips" aria-label="Copy heading and permalink for Table 1.2.7: Probability Distribution for Two Fair Coin Flips">🔗</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"><span class="process-math">\(HH\)</span></td>
<td class="c m b0 r0 l0 t0 lines">0.25</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(HT\)</span></td>
<td class="c m b0 r0 l0 t0 lines">0.25</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">0.25</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(TT\)</span></td>
<td class="c m b0 r0 l0 t0 lines">0.25</td>
</tr>
</table></div></figure><div class="autopermalink" data-description="Answer 1.2.2.b.1"><a href="#exercises-Probability-2-3-2" title="Copy heading and permalink for Answer 1.2.2.b.1" aria-label="Copy heading and permalink for Answer 1.2.2.b.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 1.2.2.b"><a href="#exercises-Probability-2-3" title="Copy heading and permalink for Task 1.2.2.b" aria-label="Copy heading and permalink for Task 1.2.2.b">🔗</a></div></article><article class="task exercise-like" id="exercises-Probability-2-4"><h5 class="heading"><span class="codenumber">(c)</span></h5>
<div class="para" id="exercises-Probability-2-4-1-1">Make a probability distribution table assuming the coin comes up heads with probability 0.3.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-2-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="exercises-Probability-2-4-2" 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">
<figure class="table table-like" id="exercises-Probability-2-4-2-1"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.2.8<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Probability Distribution for Two Fair Coin Flips<div class="autopermalink" data-description="Table 1.2.8: Probability Distribution for Two Fair Coin Flips"><a href="#exercises-Probability-2-4-2-1" title="Copy heading and permalink for Table 1.2.8: Probability Distribution for Two Fair Coin Flips" aria-label="Copy heading and permalink for Table 1.2.8: Probability Distribution for Two Fair Coin Flips">🔗</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"><span class="process-math">\(HH\)</span></td>
<td class="c m b0 r0 l0 t0 lines">0.09</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(HT\)</span></td>
<td class="c m b0 r0 l0 t0 lines">0.21</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">0.21</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(TT\)</span></td>
<td class="c m b0 r0 l0 t0 lines">0.49</td>
</tr>
</table></div></figure><div class="autopermalink" data-description="Answer 1.2.2.c.1"><a href="#exercises-Probability-2-4-2" title="Copy heading and permalink for Answer 1.2.2.c.1" aria-label="Copy heading and permalink for Answer 1.2.2.c.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 1.2.2.c"><a href="#exercises-Probability-2-4" title="Copy heading and permalink for Task 1.2.2.c" aria-label="Copy heading and permalink for Task 1.2.2.c">🔗</a></div></article><div class="autopermalink" data-description="Exercise 1.2.2"><a href="#exercises-Probability-2" title="Copy heading and permalink for Exercise 1.2.2" aria-label="Copy heading and permalink for Exercise 1.2.2">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Probability-3"><h4 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="introduction" id="exercises-Probability-3-1"><div class="para" id="exercises-Probability-3-1-1">Suppose we roll a die two times. Answer the questions below.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></div>
<article class="task exercise-like" id="exercises-Probability-3-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
<div class="para" id="exercises-Probability-3-2-1-1">Write all outcomes in the sample space <span class="process-math">\(\Omega\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-3-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="exercises-Probability-3-2-2" 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 logical" id="exercises-Probability-3-2-2-1">
<div class="displaymath process-math" id="exercises-Probability-3-2-2-1-1">
\begin{align*}
\Omega = \{\amp (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), \\
\amp (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), \\
\amp (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), \\
\amp (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), \\
\amp (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), \\
\amp (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\}
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-3-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 1.2.3.a.1"><a href="#exercises-Probability-3-2-2" title="Copy heading and permalink for Answer 1.2.3.a.1" aria-label="Copy heading and permalink for Answer 1.2.3.a.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 1.2.3.a"><a href="#exercises-Probability-3-2" title="Copy heading and permalink for Task 1.2.3.a" aria-label="Copy heading and permalink for Task 1.2.3.a">🔗</a></div></article><article class="task exercise-like" id="exercises-Probability-3-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
<div class="para" id="exercises-Probability-3-3-1-1">Make a probability distribution table for <span class="process-math">\(\Omega\)</span> assuming the die is fair.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-3-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Task 1.2.3.b"><a href="#exercises-Probability-3-3" title="Copy heading and permalink for Task 1.2.3.b" aria-label="Copy heading and permalink for Task 1.2.3.b">🔗</a></div></article><article class="task exercise-like" id="exercises-Probability-3-4"><h5 class="heading"><span class="codenumber">(c)</span></h5>
<div class="para" id="exercises-Probability-3-4-1-1">Let <span class="process-math">\(A\)</span> be the event that the second roll is higher than the first, and let <span class="process-math">\(B\)</span> be the event that the first roll is even. Find <span class="process-math">\(\Pr(A), \Pr(B)\text{,}\)</span> and <span class="process-math">\(\Pr(A \mid B)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-3-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="solutions"><details id="exercises-Probability-3-3-2" 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="exercises-Probability-3-3-2-1">We’ll avoid an overly large table and note that, since the die is fair, every outcome is equally likely. Therefore, <span class="process-math">\(\Pr(x) = \frac{1}{36}\)</span> for every <span class="process-math">\(x\in \Omega\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-3-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 1.2.3.b.1"><a href="#exercises-Probability-3-3-2" title="Copy heading and permalink for Answer 1.2.3.b.1" aria-label="Copy heading and permalink for Answer 1.2.3.b.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 1.2.3.b"><a href="#exercises-Probability-3-3" title="Copy heading and permalink for Task 1.2.3.b" aria-label="Copy heading and permalink for Task 1.2.3.b">🔗</a></div></article><article class="task exercise-like" id="exercises-Probability-3-4"><h5 class="heading"><span class="codenumber">(c)</span></h5>
<div class="para" id="exercises-Probability-3-4-1-1">Let <span class="process-math">\(A\)</span> be the event that the second roll is higher than the first, and let <span class="process-math">\(B\)</span> be the event that the first roll is even. Find <span class="process-math">\(\Pr(A), \Pr(B)\text{,}\)</span> and <span class="process-math">\(\Pr(A \cap B)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-3-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="exercises-Probability-3-4-2" 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 logical" id="exercises-Probability-3-4-2-1">
<div class="displaymath process-math" id="exercises-Probability-3-4-2-1-1">
\begin{align*}
A = \{ \amp (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), \\
\amp (2, 3), (2, 4), (2, 5), (2, 6), \\
\amp (3, 4), (3, 5), (3, 6), \\
\amp (4, 5), (4, 6), \\
\amp (5, 6)\} \\
B = \{ \amp (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), \\
\amp (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), \\
\amp (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\} \\
A \cap B = \{ \amp (2, 3), (2, 4), (2, 5), (2, 6), \\
\amp (4, 5), (4, 6)\}
\end{align*}
</div>
<div class="para">Therefore <span class="process-math">\(\Pr(A) = \frac{15}{36} = \frac{5}{12}, \Pr(B) = \frac{18}{36} = \frac{1}{2}, \Pr(A \cap B) = \frac{6}{36} = \frac{1}{6}.\)</span>
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-3-4-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 1.2.3.c.1"><a href="#exercises-Probability-3-4-2" title="Copy heading and permalink for Answer 1.2.3.c.1" aria-label="Copy heading and permalink for Answer 1.2.3.c.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 1.2.3.c"><a href="#exercises-Probability-3-4" title="Copy heading and permalink for Task 1.2.3.c" aria-label="Copy heading and permalink for Task 1.2.3.c">🔗</a></div></article><div class="autopermalink" data-description="Exercise 1.2.3"><a href="#exercises-Probability-3" title="Copy heading and permalink for Exercise 1.2.3" aria-label="Copy heading and permalink for Exercise 1.2.3">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Probability-4"><h4 class="heading"><span class="codenumber">4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Probability-4-1-1">Suppose a die has the values <span class="process-math">\(1, 2, 3, 4, 5, 6\)</span> on the faces, but the die is not fair. Instead, the probabilities scale by the same amount as the face values. For example, a result of 4 is twice as likely as a result of 2, since 4 is twice as large as 2; a result of 6 is six times more likely than a result of 1; and so on. Write a probability distribution table for this die.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-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="exercises-Probability-4-2" 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">
<figure class="table table-like" id="exercises-Probability-4-2-1"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.2.9<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Probability Distribution for a Linearly Scaled Die<div class="autopermalink" data-description="Table 1.2.9: Probability Distribution for a Linearly Scaled Die"><a href="#exercises-Probability-4-2-1" title="Copy heading and permalink for Table 1.2.9: Probability Distribution for a Linearly Scaled Die" aria-label="Copy heading and permalink for Table 1.2.9: Probability Distribution for a Linearly Scaled 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">\(1/21\)</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">\(2/21\)</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">\(3/21\)</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">\(4/21\)</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">\(5/21\)</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">\(6/21\)</span></td>
</tr>
</table></div></figure><div class="autopermalink" data-description="Answer 1.2.4.1"><a href="#exercises-Probability-4-2" title="Copy heading and permalink for Answer 1.2.4.1" aria-label="Copy heading and permalink for Answer 1.2.4.1">🔗</a></div>
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<div class="autopermalink" data-description="Exercise 1.2.4"><a href="#exercises-Probability-4" title="Copy heading and permalink for Exercise 1.2.4" aria-label="Copy heading and permalink for Exercise 1.2.4">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Probability-5"><h4 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Probability-5-1-1">Suppose a die has the values <span class="process-math">\(1, 2, 3, 4, 5, 6\)</span> on the faces, but the die is not fair. Instead, each even value has an equal probability, each odd value has an equal probability, and the even values are each twice as likely as the odd values to appear on a roll. Write a probability distribution table for this die.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-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 class="solutions"><details id="exercises-Probability-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">
<figure class="table table-like" id="exercises-Probability-5-2-1"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.2.10<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Probability Distribution for an Even-biased Die<div class="autopermalink" data-description="Table 1.2.10: Probability Distribution for an Even-biased Die"><a href="#exercises-Probability-5-2-1" title="Copy heading and permalink for Table 1.2.10: Probability Distribution for an Even-biased Die" aria-label="Copy heading and permalink for Table 1.2.10: Probability Distribution for an Even-biased 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">\(1/9\)</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">\(2/9\)</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">\(1/9\)</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">\(2/9\)</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">\(1/9\)</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">\(2/9\)</span></td>
</tr>
</table></div></figure><div class="autopermalink" data-description="Answer 1.2.5.1"><a href="#exercises-Probability-5-2" title="Copy heading and permalink for Answer 1.2.5.1" aria-label="Copy heading and permalink for Answer 1.2.5.1">🔗</a></div>
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<div class="autopermalink" data-description="Exercise 1.2.5"><a href="#exercises-Probability-5" title="Copy heading and permalink for Exercise 1.2.5" aria-label="Copy heading and permalink for Exercise 1.2.5">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Probability-6"><h4 class="heading"><span class="codenumber">6<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Probability-6-1-1">A toxin molecule inside a cell has a 0.3 probability of leaving the cell during a 1-minute period. For each value of <span class="process-math">\(n = 1, 2, 3, \dotsc\text{,}\)</span> find the probability of the toxin molecule leaving the cell during the <span class="process-math">\(n\)</span>th minute. What is the probability of the molecule leaving the cell during the first 3 minutes?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-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 class="autopermalink" data-description="Exercise 1.2.6"><a href="#exercises-Probability-6" title="Copy heading and permalink for Exercise 1.2.6" aria-label="Copy heading and permalink for Exercise 1.2.6">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Probability-7"><h4 class="heading"><span class="codenumber">7<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Probability-7-1-1">Each of 10 toxin molecules inside a cell has a 0.3 probability of leaving the cell during a 1-minute period. For each value of <span class="process-math">\(n = 1, 2, 3, \dotsc\text{,}\)</span> and for each value of <span class="process-math">\(0\leq k \leq n\text{,}\)</span> find the probability that exactly <span class="process-math">\(k\)</span> toxin molecules remain in the cell after the <span class="process-math">\(n\)</span>th minute.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="solutions"><details id="exercises-Probability-6-2" 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="exercises-Probability-6-2-1">For short, write <span class="process-math">\(\Pr(n)\)</span> to mean the probability of the toxin molecule leaving during the <span class="process-math">\(n\)</span>th minute. Then <span class="process-math">\(\Pr(n) = (0.7)^{n - 1} (0.3).\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-6-2-1" 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="Exercise 1.2.7"><a href="#exercises-Probability-7" title="Copy heading and permalink for Exercise 1.2.7" aria-label="Copy heading and permalink for Exercise 1.2.7">🔗</a></div></article><div class="autopermalink" data-description="Exercises 1.2"><a href="#exercises-Probability" title="Copy heading and permalink for Exercises 1.2" aria-label="Copy heading and permalink for Exercises 1.2">🔗</a></div></section><div class="autopermalink" data-description="Section 1.2: Definition of Probability"><a href="#sec-Probability" title="Copy heading and permalink for Section 1.2: Definition of Probability" aria-label="Copy heading and permalink for Section 1.2: Definition of Probability">🔗</a></div></section></div>
<div class="para" id="exercises-Probability-6-2-2">The probability of leaving during the first 3 minutes is <span class="process-math">\(\Pr(1) + \Pr(2) + \Pr(3) = 0.657.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-6-2-2" 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="Answer 1.2.6.1"><a href="#exercises-Probability-6-2" title="Copy heading and permalink for Answer 1.2.6.1" aria-label="Copy heading and permalink for Answer 1.2.6.1">🔗</a></div>
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<div class="autopermalink" data-description="Exercise 1.2.6"><a href="#exercises-Probability-6" title="Copy heading and permalink for Exercise 1.2.6" aria-label="Copy heading and permalink for Exercise 1.2.6">🔗</a></div></article><div class="autopermalink" data-description="Exercises 1.2"><a href="#exercises-Probability" title="Copy heading and permalink for Exercises 1.2" aria-label="Copy heading and permalink for Exercises 1.2">🔗</a></div></section><div class="autopermalink" data-description="Section 1.2: Definition of Probability"><a href="#sec-Probability" title="Copy heading and permalink for Section 1.2: Definition of Probability" aria-label="Copy heading and permalink for Section 1.2: Definition of Probability">🔗</a></div></section></div>
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