Boxes paradox

2026-03-12 14:58:10 -04:00
parent 463bfb0d3f
commit c0cd582019
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 <?xml version="1.0" encoding="UTF-8"?>
 <!-- When creating a new activity, make a copy of this file with appropriate name -->
 <worksheet xml:id="boxes">
  <title>Boxes</title>
  <!-- Optional introduction -->
  <introduction>
    <p>
      The following is not for credit and will not be covered on any quiz or exam.
      This is just for fun.
    </p>
  </introduction>
  <page>
    <!-- Exercises start here. -->
    <exercise>
      <introduction>
        <p>
          A superintelligent robot named Bob waits in a room with two boxes.
          Box 1 contains $1000.
          Box 2 is a mystery box containing a secret amount of money.
          Bob will offer you the choice:
          <ol>
            <li>
              <p>
                You can take the mystery box alone, or
              </p>
            </li>
            <li>
              <p>
                You can take both boxes.
              </p>
            </li>
          </ol>
          Bob is very accurate at predicting ahead of time what people will do.
          If Bob thinks that you'll choose to take only the mystery box, he'll put $1 million dollars in the mystery box.
          If Bob thinks that you'll choose to take both boxes, he'll put nothing in the mystery box.
          Bob made his prediction and placed $1 million or $0 in the mystery box before you entered the rooms and learned the rules of the game.
        </p>
        <p>
          Would you take just the mystery box? Or would you take both boxes?
        </p>
      </introduction>
      <task workspace="1in">
        <statement>
          <p>
            Let <m>\alpha</m> be the probability that Bob accurately predicts your choice.
            Write an expected value formula for the strategy of choosing just the mystery box in terms of <m>\alpha</m>.
            Write an expected value formula for the strategy of choosing both boxes in terms of <m>\alpha</m>.
            Which strategy has a higher expected value?
          </p>
        </statement>
      </task>
      <task workspace="1in">
        <statement>
          <p>
            Let <m>\beta</m> be the probability that Bob predicts that you'll take both boxes.
            Write an expected value formula for the strategy of choosing just the mystery box in terms of <m>\beta</m>.
            Write an expected value formula for the strategy of choosing both boxes in terms of <m>\beta</m>.
            Which strategy has a higher expected value?
          </p>
        </statement>
      </task>
    </exercise>
  </page>
 </worksheet>
@@ -62,7 +62,14 @@
      <title>Exam Review</title>
      <xi:include href="./review/Exam-1-Review.ptx"/>
    </chapter>
    <chapter xml:id="fun">
      <title>Just for Fun</title>
      <xi:include href="./fun/boxes.ptx"/>
    </chapter>