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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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<div class="para" id="boxes-3-1-1-1-1-2-1">You can take both boxes.<div class="autopermalink" data-description="Paragraph"><a href="#boxes-3-1-1-1-1-2-1" 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="Item"><a href="#boxes-3-1-1-1-1-2" title="Copy heading and permalink for Item" aria-label="Copy heading and permalink for Item">🔗</a></div>
</li>
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<div class="para">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 either $1 million or $0 in the mystery box before you entered the room. Would you take just the mystery box? Or would you take both boxes?</div>
<div class="autopermalink" data-description="Paragraph"><a href="#boxes-3-1-1-1" 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="boxes-3-1-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
<div class="para" id="boxes-3-1-2-1-1">Let <span class="process-math">\(\alpha\)</span> 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 <span class="process-math">\(\alpha\text{.}\)</span> Write an expected value formula for the strategy of choosing both boxes in terms of <span class="process-math">\(\alpha\text{.}\)</span> Which strategy has a higher expected value?<div class="autopermalink" data-description="Paragraph"><a href="#boxes-3-1-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="workspace" data-space="1in"></div>
<div class="autopermalink" data-description="Task 1.a"><a href="#boxes-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="boxes-3-1-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
<div class="para" id="boxes-3-1-3-1-1">Let <span class="process-math">\(\beta\)</span> 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 <span class="process-math">\(\beta\text{.}\)</span> Write an expected value formula for the strategy of choosing both boxes in terms of <span class="process-math">\(\beta\text{.}\)</span> Which strategy has a higher expected value?<div class="autopermalink" data-description="Paragraph"><a href="#boxes-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="workspace" data-space="1in"></div>
<div class="autopermalink" data-description="Task 1.b"><a href="#boxes-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><div class="autopermalink" data-description="Worksheet Exercise 1"><a href="#boxes-3-1" title="Copy heading and permalink for Worksheet Exercise 1" aria-label="Copy heading and permalink for Worksheet Exercise 1">🔗</a></div></article></section><div class="autopermalink" data-description="Worksheet: Boxes"><a href="#boxes" title="Copy heading and permalink for Worksheet: Boxes" aria-label="Copy heading and permalink for Worksheet: Boxes">🔗</a></div></section>
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