Fall-2026-Math-1041/source/quizzes/quiz-01.ptx

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<!-- When creating a new activity, make a copy of this file with appropriate name -->
<worksheet xml:id="quiz-01">
  <title>Quiz 1</title>

  <!-- Optional introduction -->
  <introduction>
    <p>
      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.
    </p>

    <p>
      Show all work unless instructed otherwise.
    </p>
  </introduction>


  <page>
    <!-- Exercises start here. -->
    <exercise>
      <introduction>
        <p>
          Write either True or False for each of the following statements.
          No justification is required.
        </p>
      </introduction>


      <task>
        <statement>
          <p>
            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>.
          </p>
        </statement>

        <solution>
          <p>
            True.
          </p>
        </solution>
      </task>


      <task>
        <statement>
          <p>
            If <m>A</m> and <m>B</m> are any events, then <m>\Pr(A \mid B) = 1 - \Pr(B \mid A)</m>.
          </p>
        </statement>

        <solution>
          <p>
            False.
          </p>
        </solution>
      </task>


      <task>
        <statement>
          <p>
            If <m>A</m> and <m>B</m> are any events, then <m>\Pr(A\cap B)\Pr(B) = \Pr(A \mid B)</m>.
          </p>
        </statement>

        <solution>
          <p>
            False.
          </p>
        </solution>
      </task>
    </exercise>

    <exercise>
      <introduction>
        <p>
          We find a 4-sided die with faces 0, 1, 3, and 5.
          An experiment consists of rolling the die two times.
        </p>
      </introduction>


      <task>
        <statement>
          <p>
            Write down the sample space <m>\Omega</m> of all possible outcomes for this experiment.
          </p>
        </statement>

        <solution>
          <p>
            <md>
              <mrow> \Omega = \{ \amp (0, 0), (0, 1), (0, 3), (0, 5), </mrow>
              <mrow> \amp (1, 0), (1, 1), (1, 3), (1, 5), </mrow>
              <mrow> \amp (3, 0), (3, 1), (3, 3), (3, 5), </mrow>
              <mrow> \amp (5, 0), (5, 1), (5, 3), (5, 5)\} </mrow>
            </md>
          </p>
        </solution>
      </task>


      <task>
        <statement>
          <p>
            Let <m>A</m> 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 <m>A</m> and calculate <m>\Pr(A)</m>.
            List the outcomes in <m>B</m> and calculate <m>\Pr(B)</m>.
          </p>
        </statement>

        <solution>
          <p>
            <md>
              <mrow> A \amp = \{(0, 0), (0, 1), (0, 3), (0, 5), (1, 3), (1, 5)\} </mrow>
              <mrow> B \amp = \{ (0, 1), (0, 3), (0, 5), (1, 0), (3, 0), (5, 0) \} </mrow>
            </md>
            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>.
          </p>
        </solution>
      </task>
    </exercise>
  </page>


  <page>
    <exercise>
      <statement>
        <p>
          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.
        </p>

        <p>
          For a patient who receives a positive test, what is the probability they have the disease?
        </p>
      </statement>

      <solution>
        <p>
          Let <m>P</m> be the event of receiving a positive test result and <m>D</m> be the event of having the disease.
          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>.
          Then, using Bayes' Theorem:
          <md>
            <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>
            <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>
            <mrow> \amp = \frac{(0.62)(0.013)}{(0.62)(0.013) + (1 - 0.994)(1 - 0.013)}  </mrow>
            <mrow> \amp \approx \boxed{0.576}  </mrow>
          </md>
        </p>
      </solution>
    </exercise>
  </page>
</worksheet>