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<div class="toc-title-box"><a href="ch-Expected-Value.html" class="internal"><span class="codenumber">3</span> <span class="title">Expected Value and Variance</span></a></div>
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<div class="toc-title-box"><a href="sec-Expected-Value.html" class="internal"><span class="codenumber">3.1</span> <span class="title">Expected Value</span></a></div>
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<div class="toc-title-box"><a href="sec-Confidence-Intervals.html" class="internal"><span class="codenumber">4.3</span> <span class="title">Confidence Intervals</span></a></div>
<ul id="ptx-toc-group-sec-Confidence-Intervals" 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">4.3</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 id="ptx-toc-group-ch-Hypothesis-Testing" class="structural toc-item-list">
<li class="toc-item toc-introduction"><div class="toc-title-box"><a href="ch-Hypothesis-Testing-2.html" class="internal"><span class="title">Introduction</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 id="ptx-toc-group-sec-One-Sample-Tests" 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 id="ptx-toc-group-sec-Two-Sample-Tests" 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 id="ptx-toc-group-sec-Power" 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 id="ptx-toc-group-sec-Chi-Squared" 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>
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</ul>
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<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 id="ptx-toc-group-ch-Linear-Regression" class="structural toc-item-list">
<li class="toc-item toc-introduction"><div class="toc-title-box"><a href="ch-Linear-Regression-2.html" class="internal"><span class="title">Introduction</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 id="ptx-toc-group-sec-Correlation" 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 id="ptx-toc-group-sec-Linear-Regression" 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>
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</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>
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<li class="toc-item toc-appendix">
<div class="toc-title-box"><a href="backmatter-2.html" class="internal"><span class="codenumber">A</span> <span class="title"><span class="process-math">\(\Phi(z)\)</span> Table</span></a></div>
<ul id="ptx-toc-group-backmatter-2" class="structural toc-item-list"><li class="toc-item toc-section"><div class="toc-title-box"><a href="app-Phi-table.html" class="internal"><span class="codenumber">A.1</span> <span class="title"><span class="process-math">\(\Phi(z)\)</span> Table of Values</span></a></div></li></ul>
</li>
<li class="toc-item toc-appendix">
<div class="toc-title-box"><a href="backmatter-3.html" class="internal"><span class="codenumber">B</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Critical Values Table</span></a></div>
<ul id="ptx-toc-group-backmatter-3" class="structural toc-item-list"><li class="toc-item toc-section"><div class="toc-title-box"><a href="app-Chi-squared-table.html" class="internal"><span class="codenumber">B.1</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Critical Values</span></a></div></li></ul>
</li>
<li class="toc-item toc-colophon"><div class="toc-title-box"><a href="backmatter-4.html" class="internal"><span class="title">Colophon</span></a></div></li>
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<main class="ptx-main"><div id="ptx-content" class="ptx-content">
<section class="section" id="sec-Conditional-Probability">
<h1 class="heading hide-type">
<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.3</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Conditional Probability</span>
</h1>
<section class="subsection" id="subsec-conditional-probability">
<h2 class="heading hide-type">
<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.3.1</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Conditional Probability</span>
</h2>
<div class="para" id="subsec-conditional-probability-2">Sometimes events can interact with each other. We would like to have the language to talk about a scenario in which evidence that one event has occurred can alter our understanding of the probability of another event occurring. We would also like to develop the mathematical tools to quantify exactly how much that probability changes.<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#subsec-conditional-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-conditional-probability"><h3 class="heading">
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.3.1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h3>
<div class="para logical" id="def-conditional-probability-1-1">
<div class="para">Let <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> be events. The <dfn class="terminology">conditional probability</dfn> of <span class="process-math">\(A\)</span> given <span class="process-math">\(B\)</span> is</div>
<div class="displaymath process-math" id="def-conditional-probability-1-1-6">
\begin{gather*}
\Pr(A \mid B) = \frac{\Pr(A\cap B)}{\Pr(B)}.
\end{gather*}
</div>
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</div>
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</article>
<article class="example example-like" id="example-rolls-conditional"><h3 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.3.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h3>
<div class="para logical" id="example-rolls-conditional-1-1">
<div class="para">An experiment consists of rolling a fair 6-sided die two times. Let <span class="process-math">\(A\)</span> be the event that the sum of the rolls is at least 10. To find <span class="process-math">\(\Pr(A)\text{,}\)</span> we note that <span class="process-math">\(A = \{(4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)\}\text{,}\)</span> which has 6 elements. Since <span class="process-math">\(|\Omega| = 36\text{,}\)</span> we have</div>
<div class="displaymath process-math" id="example-rolls-conditional-1-1-5">
\begin{gather*}
\Pr(A) = \frac{|A|}{|\Omega|} = \frac{6}{36} = \frac{1}{6}\text{.}
\end{gather*}
</div>
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#example-rolls-conditional-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="para logical" id="example-rolls-conditional-1-2">
<div class="para">Let <span class="process-math">\(B\)</span> be the event that the first roll is 6. If we pause after the first die roll seeing the value of 6, we might be more inclined to expect a sum of at least 10. The evidence that we’ve already seen changes our understanding of the situation. Since <span class="process-math">\(|B| = 6\text{,}\)</span> <span class="process-math">\(\Pr(B) = \frac{6}{36} = \frac{1}{6}\text{.}\)</span> Also, <span class="process-math">\(A\cap B = \{(6, 4), (6, 5), (6, 6)\}\text{,}\)</span> so <span class="process-math">\(\Pr(A\cap B) = \frac{3}{36} = \frac{1}{12}\text{.}\)</span> Finally:</div>
<div class="displaymath process-math" id="example-rolls-conditional-1-2-6">
\begin{gather*}
\Pr(A \mid B) = \frac{\Pr(A\cap B)}{\Pr(B)} = \frac{1/12}{1/6} = \frac{1}{2}
\end{gather*}
</div>
<div class="para">Before the experiment, we would have said there was only a <span class="process-math">\(\frac{1}{6}\)</span> chance that the sum of the rolls is at least 10. However, with the additional knowledge of seeing the first roll of 6, we find the probability of a sum of at least 10 to be <span class="process-math">\(\frac{1}{2}\text{,}\)</span> substantially more likely than before.</div>
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</div>
<div class="autopermalink" aria-hidden="true" data-description="Example 1.3.2"><a tabindex="-1" href="#example-rolls-conditional" title="Copy heading and permalink for Example 1.3.2" aria-label="Copy heading and permalink for Example 1.3.2">🔗</a></div>
</article><div class="autopermalink" aria-hidden="true" data-description="Subsection 1.3.1: Conditional Probability"><a tabindex="-1" href="#subsec-conditional-probability" title="Copy heading and permalink for Subsection 1.3.1: Conditional Probability" aria-label="Copy heading and permalink for Subsection 1.3.1: Conditional Probability">🔗</a></div>
</section>
<section class="subsection" id="subsec-diagnostic-testing">
<h2 class="heading hide-type">
<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.3.2</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Diagnostic Testing</span>
</h2>
<div class="para" id="subsec-diagnostic-testing-2">Diagnostic tests for diseases aren’t perfect. When a test comes back positive or negative, a patient will want to understand the (conditional) probability that they have or don’t have the disease based on the evidence (the diagnostic test result).<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#subsec-diagnostic-testing-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<article class="definition definition-like" id="def-sensitivity-specificity"><h3 class="heading">
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.3.3</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h3>
<div class="para" id="def-sensitivity-specificity-1-1">The <dfn class="terminology">sensitivity</dfn> of a diagnostic test is the probability that a patient who has the disease will see a positive test result. The <dfn class="terminology">specificity</dfn> is the probability that a patient who does not have the disease will see a negative test result.<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#def-sensitivity-specificity-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" aria-hidden="true" data-description="Definition 1.3.3"><a tabindex="-1" href="#def-sensitivity-specificity" title="Copy heading and permalink for Definition 1.3.3" aria-label="Copy heading and permalink for Definition 1.3.3">🔗</a></div>
</article>
<div class="para" id="subsec-diagnostic-testing-4">Introducing event notation, let <span class="process-math">\(D\)</span> be the event that a patient has the disease, and let <span class="process-math">\(P\)</span> be the event that they receive a positive test result. Then the sensitivity of the diagnostic test is <span class="process-math">\(\Pr(P \mid D)\text{,}\)</span> and the specificity is <span class="process-math">\(\Pr(P^c \mid D^c)\text{.}\)</span> However, when the patient takes a diagnostic test, the conditional probabilities they would be most interested in would be <span class="process-math">\(\Pr(D \mid P)\)</span> and <span class="process-math">\(\Pr(D^c \mid P^c)\text{.}\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#subsec-diagnostic-testing-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="para" id="subsec-diagnostic-testing-5">Bayes’ Theorem expresses the relationship between a conditional probability <span class="process-math">\(\Pr(A \mid B)\)</span> and the flipped conditional probability <span class="process-math">\(\Pr(B \mid A)\text{.}\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#subsec-diagnostic-testing-5" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<article class="theorem theorem-like" id="thm-Bayes-v1"><h3 class="heading">
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.3.4</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>
</h3>
<div class="para logical" id="thm-Bayes-v1-2-1">
<div class="displaymath process-math" id="thm-Bayes-v1-2-1-1">
\begin{gather*}
\Pr(B \mid A) = \frac{\Pr(A \mid B)\Pr(B)}{\Pr(A)}
\end{gather*}
</div>
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#thm-Bayes-v1-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
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</article>
<div class="para logical" id="subsec-diagnostic-testing-7">
<div class="para">For example, if a patient sees a positive diagnostic test result, they might try to calculate:</div>
<div class="displaymath process-math" id="subsec-diagnostic-testing-7-1">
\begin{gather*}
\Pr(D \mid P) = \frac{\Pr(P \mid D)\Pr(D)}{\Pr(P)}
\end{gather*}
</div>
<div class="para">It will take some work to be able to use this formula. We likely don’t have direct access to <span class="process-math">\(\Pr(P)\)</span> or <span class="process-math">\(\Pr(D)\text{.}\)</span>
</div>
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</div>
<div class="para logical" id="subsec-diagnostic-testing-8">
<div class="para">Let’s consider <span class="process-math">\(\Pr(P)\text{,}\)</span> the probability of receiving a positive test result. The sensitivity <span class="process-math">\(\Pr(P \mid D)\)</span> tells us this probability under the condition that the patient has the disease. For a patient who doesn’t have the disease, the specificity isn’t quite the number we’re looking for. However, consider the complementary probability:</div>
<div class="displaymath process-math" id="subsec-diagnostic-testing-8-3">
\begin{gather*}
\Pr(P \mid D^c) = 1 - \Pr(P^c \mid D^c)
\end{gather*}
</div>
<div class="para">The total <span class="process-math">\(\Pr(P)\)</span> can be divided into two categories: patients who have the disease and test positive, and patients who don’t have the disease and test positive. So:</div>
<div class="displaymath process-math" id="subsec-diagnostic-testing-8-5">
\begin{align*}
\Pr(P) \amp = \Pr(P\cap D) + \Pr(P\cap D^c) \\
\amp = \Pr(P\mid D)\Pr(D) + \Pr(P \mid D^c)\Pr(D^c) \\
\amp = \Pr(P\mid D)\Pr(D) + (1 - \Pr(P^c \mid D^c))\Pr(D^c) \\
\amp = (\text{sensitivity})\Pr(D) + (1 - \text{ specificity})\Pr(D^c)
\end{align*}
</div>
<div class="para">We can take this breakdown of <span class="process-math">\(\Pr(P)\)</span> and write a new version of Bayes’ Theorem:</div>
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</div>
<article class="theorem theorem-like" id="thm-Bayes-v2"><h3 class="heading">
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.3.5</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>
</h3>
<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) + (1 - \Pr(A^c \mid B^c))\Pr(B^c)}
\end{gather*}
</div>
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</div>
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</article>
<div class="para" id="subsec-diagnostic-testing-10">We’re still missing a crucial piece of information: <span class="process-math">\(\Pr(D)\text{,}\)</span> the probability (not conditioned on any evidence) that the patient has the disease. This is often referred to as the <dfn class="terminology">prior</dfn>, as in, our prior understanding of the probability of something before we gained some new information from evidence. The conditional probability calculated using Bayes’ Theorem is usually called the <dfn class="terminology">posterior</dfn> (i.e., after taking evidence into account).<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#subsec-diagnostic-testing-10" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="para" id="subsec-diagnostic-testing-11">There isn’t always one single number that’s reasonable to use as the prior probability. For example, in a diagnostic testing situation, the <dfn class="terminology">prevalence</dfn> of the disease—i.e., the proportion of the population who have the disease—might feel like a natural number to use as the prior. However, what prevalence should you use? During the COVID-19 pandemic, the prevalence of COVID in a particular country, state, and city might be different. There’s also the possibility of applying Bayes’ Theorem multiple times to take into account multiple pieces of evidence, using the posterior probability from one application of Bayes’ Theorem to play the role of the prior probability in the next. This idea would apply if, for example, a patient took a second diagnostic test to double-check. (Complicating the issue further, the developers of diagnostic tests often publish two or even more sets of sensitivity and specificity values, depending on whether a patient is already showing certain symptoms.)<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#subsec-diagnostic-testing-11" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<article class="example example-like" id="subsec-diagnostic-testing-12"><h3 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.3.6</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h3>
<div class="para" id="subsec-diagnostic-testing-12-1-1">A 50-year old woman with no symptoms is screened for breast cancer and tests positive. If the prevalence of breast cancer for women in her age group is 1% and the particular screening process used has a sensitivity of 90% and a specificity of 91%, what is the probability that the woman has breast cancer given her positive result?<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#subsec-diagnostic-testing-12-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions">
<details id="subsec-diagnostic-testing-12-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary>
<div class="solution solution-like knowl__content">
<div class="para logical" id="subsec-diagnostic-testing-12-2-1">
<div class="para">Let <span class="process-math">\(P\)</span> be the event of testing positive and <span class="process-math">\(D\)</span> the event of having the disease. Then the prevalence <span class="process-math">\(\Pr(D)\)</span> is given as 1%, or 0.01. The sensitivity is <span class="process-math">\(\Pr(P\mid D) = 0.9\text{,}\)</span> and the specificity is <span class="process-math">\(\Pr(P^c\mid D^c) = 0.91\text{.}\)</span> So, according to Bayes’ Theorem:</div>
<div class="displaymath process-math" id="subsec-diagnostic-testing-12-2-1-6">
\begin{align*}
\Pr(D\mid P) \amp = \frac{\Pr(P\mid D)\Pr(D)}{\Pr(P\mid D)\Pr(D) + (1 - \Pr(P^c\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">This may seem like a surprising result. Despite sensitivity and specificty values around 90%, it turns out a positive test result only indicates a less than 10% chance of actually having the disease. Keep in mind that many ideas in probability and statistics can be highly counterintuitive. It’s important to be very precise with statements and calculations.</div>
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</section>
<section class="exercises" id="exercises-Conditional-Probability">
<h2 class="heading hide-type">
<span class="type">Exercises</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.3.3</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Exercises</span>
</h2>
<div class="exercisegroup" id="exercises-Conditional-Probability-1">
<h3 class="heading"><span class="title">Exercise Group.</span></h3>
<div class="introduction" id="exercises-Conditional-Probability-1-1">
<div class="para" id="exercises-Conditional-Probability-1-1-1">In each of the following scenarios with given events <span class="process-math">\(A\)</span> and <span class="process-math">\(B\text{,}\)</span> alculate <span class="process-math">\(\Pr(A), \Pr(B)\text{,}\)</span> <span class="process-math">\(\Pr(A\cap B)\text{,}\)</span> <span class="process-math">\(\Pr(A \mid B)\text{,}\)</span> and <span class="process-math">\(\Pr(B \mid A)\text{.}\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Conditional-Probability-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="exercisegroup-exercises">
<article class="exercise exercise-like" id="exercises-Conditional-Probability-1-2"><h4 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Conditional-Probability-1-2-1-1">An experiment consists of rolling a fair die two times. Let <span class="process-math">\(A\)</span> be the event that the sum is even, and let <span class="process-math">\(B\)</span> be the event that the second roll is higher than the first.<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Conditional-Probability-1-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="solutions">
<details id="exercises-Conditional-Probability-1-2-2" 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-Conditional-Probability-1-2-2-1">
<div class="displaymath process-math" id="exercises-Conditional-Probability-1-2-2-1-1">
\begin{align*}
A = \{ \amp (1, 1), (1, 3), (1, 5), (2, 2), (2, 4), (2, 6), \\
\amp (3, 1), (3, 3), (3, 5), (4, 2), (4, 4), (4, 6), \\
\amp (5, 1), (5, 3), (5, 5), (6, 2), (6, 4), (6, 6)\} \\
B = \{ \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)\} \\
A \cap B = \{ \amp (1, 3), (1, 5), (2, 4), (2, 6), (3, 5), (4, 6)\}
\end{align*}
</div>
<div class="para">So <span class="process-math">\(\Pr(A) = \frac{18}{36} = \frac{1}{2}\text{,}\)</span> <span class="process-math">\(\Pr(B) = \frac{15}{36} = \frac{5}{12}\text{,}\)</span> and <span class="process-math">\(\Pr(A\cap B) = \frac{6}{36} = \frac{1}{6}\text{.}\)</span> Finally:</div>
<div class="displaymath process-math" id="exercises-Conditional-Probability-1-2-2-1-5">
\begin{align*}
\Pr(A \mid B) \amp = \frac{\Pr(A\cap B)}{\Pr(B)} = \frac{6/36}{15/36} = \frac{6}{15} = \frac{2}{5} \\
\Pr(B \mid A) \amp = \frac{\Pr(B\cap A)}{\Pr(A)} = \frac{6/36}{18/36} = \frac{6}{18} = \frac{1}{3}
\end{align*}
</div>
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<article class="exercise exercise-like" id="exercises-Conditional-Probability-1-3"><h4 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Conditional-Probability-1-3-1-1">An experiment consists of flipping a fair coin three times. Let <span class="process-math">\(A\)</span> be the event that the first and second flips match. Let <span class="process-math">\(B\)</span> be the event that there are at least two heads.<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Conditional-Probability-1-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="solutions">
<details id="exercises-Conditional-Probability-1-3-2" 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 logical" id="exercises-Conditional-Probability-1-3-2-1">
<div class="displaymath process-math" id="exercises-Conditional-Probability-1-3-2-1-1">
\begin{align*}
A \amp = \{ HHH, HHT, TTH, TTT \} \\
B \amp = \{ HHH, HHT, HTH, THH \} \\
A\cap B \amp = \{HHH, HHT\}
\end{align*}
</div>
<div class="para">So <span class="process-math">\(\Pr(A) = \frac{4}{8} = \frac{1}{2}\text{,}\)</span> <span class="process-math">\(\Pr(B) = \frac{4}{8} = \frac{1}{2}\text{,}\)</span> and <span class="process-math">\(\Pr(A\cap B) = \frac{2}{8} = \frac{1}{4}\text{.}\)</span> Finally:</div>
<div class="displaymath process-math" id="exercises-Conditional-Probability-1-3-2-1-5">
\begin{align*}
\Pr(A \mid B) \amp = \frac{\Pr(A\cap B)}{\Pr(B)} = \frac{2/8}{4/8} = \frac{2}{4} = \frac{1}{2} \\
\Pr(B \mid A) \amp = \frac{\Pr(B\cap A)}{\Pr(A)} = \frac{2/8}{4/8} = \frac{2}{4} = \frac{1}{2}
\end{align*}
</div>
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<article class="exercise exercise-like" id="exercises-Conditional-Probability-2"><h3 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
<div class="introduction" id="exercises-Conditional-Probability-2-1">
<div class="para" id="exercises-Conditional-Probability-2-1-1">A diagnostic test is developed to detect a disease present in 3.2% of the population. For a patient who has the disease, the test will accurately give a positive result 65% of the time. When the patient does not have the disease, the test will accurately give a negative result 99.9% of the time.<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-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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<article class="task exercise-like" id="exercises-Conditional-Probability-2-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
<div class="para" id="exercises-Conditional-Probability-2-2-1-1">For a patient who receives a positive test, what is the probability they have the disease?<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Conditional-Probability-2-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="solutions">
<details id="exercises-Conditional-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 logical" id="exercises-Conditional-Probability-2-2-2-1">
<div class="para">Let <span class="process-math">\(P\)</span> be the event of testing positive and <span class="process-math">\(D\)</span> the event of having the disease. Then the prevalence <span class="process-math">\(\Pr(D)\)</span> is given as 3.2%, or 0.032. The sensitivity is <span class="process-math">\(\Pr(P\mid D) = 0.65\text{,}\)</span> and the specificity is <span class="process-math">\(\Pr(P^c\mid D^c) = 0.999\text{.}\)</span> So, according to Bayes’ Theorem:</div>
<div class="displaymath process-math" id="exercises-Conditional-Probability-2-2-2-1-6">
\begin{align*}
\Pr(D\mid P) \amp = \frac{\Pr(P\mid D)\Pr(D)}{\Pr(P\mid D)\Pr(D) + (1 - \Pr(P^c\mid D^c))\Pr(D^c)} \\
\amp = \frac{(0.65)(0.032)}{(0.65)(0.032) + (1 - 0.999)(1 - 0.032)} \\
\amp \approx 0.96
\end{align*}
</div>
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<article class="task exercise-like" id="exercises-Conditional-Probability-2-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
<div class="para" id="exercises-Conditional-Probability-2-3-1-1">For a patient who receives a negative test, what is the probability they do not have the disease?<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Conditional-Probability-2-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="solutions">
<details id="exercises-Conditional-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">
<div class="para logical" id="exercises-Conditional-Probability-2-3-2-1">
<div class="displaymath process-math" id="exercises-Conditional-Probability-2-3-2-1-1">
\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) + (1 - \Pr(P\mid D))\Pr(D)} \\
\amp = \frac{(0.999)(1 - 0.032)}{(0.999)(1 - 0.032) + (1 - 0.65)(0.032)} \\
\amp \approx 0.99
\end{align*}
</div>
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