Quiz 1 solutions
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<xi:include href="./notes/week01.ptx" />
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<xi:include href="./notes/week01.ptx" />
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</chapter>
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</chapter>
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<chapter xml:id="quizzes">
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<title>Quizzes</title>
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<xi:include href="./quizzes/quiz-01.ptx"/>
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</chapter>
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<chapter xml:id="recitations">
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<chapter xml:id="recitations">
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<title>Recitations</title>
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<title>Recitations</title>
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<?xml version="1.0" encoding="UTF-8"?>
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<!-- When creating a new activity, make a copy of this file with appropriate name -->
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<worksheet xml:id="quiz-01">
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<title>Quiz 1</title>
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<!-- Optional introduction -->
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<introduction>
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<p>
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The following work should be completed individually.
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Use of notes or textbooks is not allowed.
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You may use a scientific calculator, not a graphing calculator or phone app.
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</p>
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<p>
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Show all work unless instructed otherwise.
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</p>
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</introduction>
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<page>
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<!-- Exercises start here. -->
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<exercise>
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<introduction>
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<p>
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Write either True or False for each of the following statements.
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No justification is required.
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</p>
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</introduction>
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<task>
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<statement>
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<p>
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If <m>A</m> and <m>B</m> are any sets, then <m>|A\cap B| \leq |A|</m> and <m>|A\cap B| \leq |B|</m>.
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</p>
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</statement>
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<solution>
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<p>
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True.
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</p>
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</solution>
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</task>
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<task>
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<statement>
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<p>
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If <m>A</m> and <m>B</m> are any events, then <m>\Pr(A \mid B) = 1 - \Pr(B \mid A)</m>.
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</p>
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</statement>
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<solution>
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<p>
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False.
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</p>
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</solution>
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</task>
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<task>
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<statement>
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<p>
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If <m>A</m> and <m>B</m> are any events, then <m>\Pr(A\cap B)\Pr(B) = \Pr(A \mid B)</m>.
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</p>
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</statement>
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<solution>
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<p>
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False.
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</p>
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</solution>
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</task>
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</exercise>
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<exercise>
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<introduction>
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<p>
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We find a 4-sided die with faces 0, 1, 3, and 5.
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An experiment consists of rolling the die two times.
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</p>
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</introduction>
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<task>
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<statement>
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<p>
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Write down the sample space <m>\Omega</m> of all possible outcomes for this experiment.
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</p>
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</statement>
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<solution>
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<p>
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<md>
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<mrow> \Omega = \{ \amp (0, 0), (0, 1), (0, 3), (0, 5), </mrow>
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<mrow> \amp (1, 0), (1, 1), (1, 3), (1, 5), </mrow>
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<mrow> \amp (3, 0), (3, 1), (3, 3), (3, 5), </mrow>
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<mrow> \amp (5, 0), (5, 1), (5, 3), (5, 5)\} </mrow>
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</md>
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</p>
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</solution>
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</task>
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<task>
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<statement>
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<p>
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Let <m>A</m> be the event that the second roll is at least twice the value of the first roll.
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Let m B be the event that the sum of the rolls is odd.
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Assume the die is fair.
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List the outcomes in <m>A</m> and calculate <m>\Pr(A)</m>.
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List the outcomes in <m>B</m> and calculate <m>\Pr(B)</m>.
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</p>
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</statement>
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<solution>
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<p>
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<md>
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<mrow> A \amp = \{(0, 0), (0, 1), (0, 3), (0, 5), (1, 3), (1, 5)\} </mrow>
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<mrow> B \amp = \{ (0, 1), (0, 3), (0, 5), (1, 0), (3, 0), (5, 0) \} </mrow>
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</md>
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Then <m>\Pr(A) = \frac{|A|}{|\Omega|} = \frac{6}{16} = \frac{3}{8}</m>, and <m>\Pr(B) = \frac{|B|}{|\Omega|} = \frac{6}{16} = \frac{3}{8}</m>.
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</p>
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</solution>
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</task>
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</exercise>
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</page>
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<page>
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<exercise>
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<statement>
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<p>
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A diagnostic test is developed to detect a disease present in 1.3% of the population.
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For a patient who has the disease, the test will accurately give a positive result 62% of the time.
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When the patient does not have the disease, the test will accurately give a negative result 99.4% of the time.
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</p>
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<p>
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For a patient who receives a positive test, what is the probability they have the disease?
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</p>
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</statement>
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<solution>
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<p>
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Let <m>P</m> be the event of receiving a positive test result and <m>D</m> be the event of having the disease.
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The given information is: <m>\Pr(D) = 0.013, \Pr(P\mid D) = 0.62</m>, and <m>\Pr(P^c \mid D^c) = 0.994</m>.
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Then, using Bayes' Theorem:
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<md>
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<mrow> \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)} </mrow>
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<mrow> \amp = \frac{\Pr(P \mid D)\Pr(D)}{\Pr(P \mid D)\Pr(D) + (1 - \Pr(P^c \mid D^c))(1 - \Pr(D))} </mrow>
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<mrow> \amp = \frac{(0.62)(0.013)}{(0.62)(0.013) + (1 - 0.994)(1 - 0.013)} </mrow>
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<mrow> \amp \approx \boxed{0.576} </mrow>
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</md>
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</p>
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</solution>
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</exercise>
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</page>
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</worksheet>
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