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<h1 class="heading"><a href="root-1-2.html"><span class="title">Introduction to Probability and Statistics for the Life Sciences</span> <span class="subtitle">Spring 2026 MATH 1044</span></a></h1>
<p class="byline">Andy Eisenberg</p>
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<section class="worksheet" id="quiz-01" data-margins="0.75in 0.75in 0.75in 0.75in"><h1 class="heading hide-type">
<span class="type">Worksheet</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Quiz 1</span><div class="print-links"><a href="quiz-01-printable.html" class="print-link" title="Print preview"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe8ad;</span></a></div>
</h1>
<section class="introduction" id="quiz-01-2"><div class="para" id="quiz-01-2-1">The following work should be completed individually. Use of notes or textbooks is not allowed. You may use a scientific calculator, not a graphing calculator or phone app.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="para" id="quiz-01-2-2">Show all work unless instructed otherwise.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-2-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></section><section class="onepage" id="quiz-01-3"><article class="exercise exercise-like" id="quiz-01-3-1"><h3 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
<div class="introduction" id="quiz-01-3-1-1"><div class="para" id="quiz-01-3-1-1-1">Write either True or False for each of the following statements. No justification is required.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></div>
<article class="task exercise-like" id="quiz-01-3-1-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
<div class="para" id="quiz-01-3-1-2-1-1">If <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> are any sets, then <span class="process-math">\(|A\cap B| \leq |A|\)</span> and <span class="process-math">\(|A\cap B| \leq |B|\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-1-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="quiz-01-3-1-2-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
<div class="para" id="quiz-01-3-1-2-2-1">True.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-1-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Solution 1.a.1"><a href="#quiz-01-3-1-2-2" title="Copy heading and permalink for Solution 1.a.1" aria-label="Copy heading and permalink for Solution 1.a.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 1.a"><a href="#quiz-01-3-1-2" title="Copy heading and permalink for Task 1.a" aria-label="Copy heading and permalink for Task 1.a">🔗</a></div></article><article class="task exercise-like" id="quiz-01-3-1-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
<div class="para" id="quiz-01-3-1-3-1-1">If <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> are any events, then <span class="process-math">\(\Pr(A \mid B) = 1 - \Pr(B \mid A)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-1-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="solutions"><details id="quiz-01-3-1-3-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
<div class="para" id="quiz-01-3-1-3-2-1">False.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-1-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Solution 1.b.1"><a href="#quiz-01-3-1-3-2" title="Copy heading and permalink for Solution 1.b.1" aria-label="Copy heading and permalink for Solution 1.b.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 1.b"><a href="#quiz-01-3-1-3" title="Copy heading and permalink for Task 1.b" aria-label="Copy heading and permalink for Task 1.b">🔗</a></div></article><article class="task exercise-like" id="quiz-01-3-1-4"><h4 class="heading"><span class="codenumber">(c)</span></h4>
<div class="para" id="quiz-01-3-1-4-1-1">If <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> are any events, then <span class="process-math">\(\Pr(A\cap B)\Pr(B) = \Pr(A \mid B)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-1-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="quiz-01-3-1-4-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
<div class="para" id="quiz-01-3-1-4-2-1">False.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-1-4-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Solution 1.c.1"><a href="#quiz-01-3-1-4-2" title="Copy heading and permalink for Solution 1.c.1" aria-label="Copy heading and permalink for Solution 1.c.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 1.c"><a href="#quiz-01-3-1-4" title="Copy heading and permalink for Task 1.c" aria-label="Copy heading and permalink for Task 1.c">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 1"><a href="#quiz-01-3-1" title="Copy heading and permalink for Worksheet Exercise 1" aria-label="Copy heading and permalink for Worksheet Exercise 1">🔗</a></div></article><article class="exercise exercise-like" id="quiz-01-3-2"><h3 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
<div class="introduction" id="quiz-01-3-2-1"><div class="para" id="quiz-01-3-2-1-1">We find a 4-sided die with faces 0, 1, 3, and 5. An experiment consists of rolling the die two times.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></div>
<article class="task exercise-like" id="quiz-01-3-2-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
<div class="para" id="quiz-01-3-2-2-1-1">Write down the sample space <span class="process-math">\(\Omega\)</span> of all possible outcomes for this experiment.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-2-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="quiz-01-3-2-2-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
<div class="para logical" id="quiz-01-3-2-2-2-1">
<div class="displaymath process-math" id="quiz-01-3-2-2-2-1-1">
\begin{align*}
\Omega = \{ \amp (0, 0), (0, 1), (0, 3), (0, 5), \\
\amp (1, 0), (1, 1), (1, 3), (1, 5), \\
\amp (3, 0), (3, 1), (3, 3), (3, 5), \\
\amp (5, 0), (5, 1), (5, 3), (5, 5)\}
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-2-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Solution 2.a.1"><a href="#quiz-01-3-2-2-2" title="Copy heading and permalink for Solution 2.a.1" aria-label="Copy heading and permalink for Solution 2.a.1">🔗</a></div>
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<div class="autopermalink" data-description="Task 2.a"><a href="#quiz-01-3-2-2" title="Copy heading and permalink for Task 2.a" aria-label="Copy heading and permalink for Task 2.a">🔗</a></div></article><article class="task exercise-like" id="quiz-01-3-2-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
<div class="para" id="quiz-01-3-2-3-1-1">Let <span class="process-math">\(A\)</span> be the event that the second roll is at least twice the value of the first roll. Let m B be the event that the sum of the rolls is odd. Assume the die is fair. List the outcomes in <span class="process-math">\(A\)</span> and calculate <span class="process-math">\(\Pr(A)\text{.}\)</span> List the outcomes in <span class="process-math">\(B\)</span> and calculate <span class="process-math">\(\Pr(B)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-3-2-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="solutions"><details id="quiz-01-3-2-3-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
<div class="para logical" id="quiz-01-3-2-3-2-1">
<div class="displaymath process-math" id="quiz-01-3-2-3-2-1-1">
\begin{align*}
A \amp = \{(0, 0), (0, 1), (0, 3), (0, 5), (1, 3), (1, 5)\} \\
B \amp = \{ (0, 1), (0, 3), (0, 5), (1, 0), (3, 0), (5, 0) \}
\end{align*}
</div>
<div class="para">Then <span class="process-math">\(\Pr(A) = \frac{|A|}{|\Omega|} = \frac{6}{16} = \frac{3}{8}\text{,}\)</span> and <span class="process-math">\(\Pr(B) = \frac{|B|}{|\Omega|} = \frac{6}{16} = \frac{3}{8}\text{.}\)</span>
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<div class="autopermalink" data-description="Solution 2.b.1"><a href="#quiz-01-3-2-3-2" title="Copy heading and permalink for Solution 2.b.1" aria-label="Copy heading and permalink for Solution 2.b.1">🔗</a></div>
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<div class="autopermalink" data-description="Task 2.b"><a href="#quiz-01-3-2-3" title="Copy heading and permalink for Task 2.b" aria-label="Copy heading and permalink for Task 2.b">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 2"><a href="#quiz-01-3-2" title="Copy heading and permalink for Worksheet Exercise 2" aria-label="Copy heading and permalink for Worksheet Exercise 2">🔗</a></div></article></section><section class="onepage" id="quiz-01-4"><article class="exercise exercise-like" id="quiz-01-4-1"><h3 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
<div class="para" id="quiz-01-4-1-1-1">A diagnostic test is developed to detect a disease present in 1.3% of the population. For a patient who has the disease, the test will accurately give a positive result 62% of the time. When the patient does not have the disease, the test will accurately give a negative result 99.4% of the time.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-4-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <div class="para" id="quiz-01-4-1-1-2">For a patient who receives a positive test, what is the probability they have the disease?<div class="autopermalink" data-description="Paragraph"><a href="#quiz-01-4-1-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="solutions"><details id="quiz-01-4-1-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
<div class="para logical" id="quiz-01-4-1-2-1">
<div class="para">Let <span class="process-math">\(P\)</span> be the event of receiving a positive test result and <span class="process-math">\(D\)</span> be the event of having the disease. The given information is: <span class="process-math">\(\Pr(D) = 0.013, \Pr(P\mid D) = 0.62\text{,}\)</span> and <span class="process-math">\(\Pr(P^c \mid D^c) = 0.994\text{.}\)</span> Then, using Bayes’ Theorem:</div>
<div class="displaymath process-math" id="quiz-01-4-1-2-1-5">
\begin{align*}
\Pr(D\mid P) = \frac{\Pr(P \mid D)\Pr(D)}{\Pr(P)} \amp = \frac{\Pr(P \mid D)\Pr(D)}{\Pr(P \mid D)\Pr(D) + \Pr(P \mid D^c)\Pr(D^c)} \\
\amp = \frac{\Pr(P \mid D)\Pr(D)}{\Pr(P \mid D)\Pr(D) + (1 - \Pr(P^c \mid D^c))(1 - \Pr(D))} \\
\amp = \frac{(0.62)(0.013)}{(0.62)(0.013) + (1 - 0.994)(1 - 0.013)} \\
\amp \approx \boxed{0.576}
\end{align*}
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<div class="autopermalink" data-description="Solution 3.1"><a href="#quiz-01-4-1-2" title="Copy heading and permalink for Solution 3.1" aria-label="Copy heading and permalink for Solution 3.1">🔗</a></div>
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<div class="autopermalink" data-description="Worksheet Exercise 3"><a href="#quiz-01-4-1" title="Copy heading and permalink for Worksheet Exercise 3" aria-label="Copy heading and permalink for Worksheet Exercise 3">🔗</a></div></article></section><div class="autopermalink" data-description="Worksheet: Quiz 1"><a href="#quiz-01" title="Copy heading and permalink for Worksheet: Quiz 1" aria-label="Copy heading and permalink for Worksheet: Quiz 1">🔗</a></div></section>
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