71 lines
2.5 KiB
XML
71 lines
2.5 KiB
XML
<?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="boxes">
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<title>Boxes</title>
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<!-- Optional introduction -->
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<introduction>
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<p>
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The following is not for credit and will not be covered on any quiz or exam.
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This is just for fun.
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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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A superintelligent robot named Bob waits in a room with two boxes.
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Box 1 contains $1000.
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Box 2 is a mystery box containing a secret amount of money.
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Bob will offer you the choice:
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<ul>
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<li>
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<p>
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You can take the mystery box alone, or
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</p>
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</li>
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<li>
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<p>
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You can take both boxes.
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</p>
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</li>
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</ul>
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Bob is very accurate at predicting ahead of time what people will do.
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If Bob thinks that you'll choose to take only the mystery box, he'll put $1 million dollars in the mystery box.
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If Bob thinks that you'll choose to take both boxes, he'll put nothing in the mystery box.
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Bob made his prediction and placed either $1 million or $0 in the mystery box before you entered the room.
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Would you take just the mystery box? Or would you take both boxes?
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</p>
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</introduction>
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<task workspace="1in">
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<statement>
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<p>
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Let <m>\alpha</m> be the probability that Bob accurately predicts your choice.
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Write an expected value formula for the strategy of choosing just the mystery box in terms of <m>\alpha</m>.
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Write an expected value formula for the strategy of choosing both boxes in terms of <m>\alpha</m>.
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Which strategy has a higher expected value?
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</p>
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</statement>
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</task>
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<task workspace="1in">
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<statement>
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<p>
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Let <m>\beta</m> be the probability that Bob predicts that you'll take both boxes.
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Write an expected value formula for the strategy of choosing just the mystery box in terms of <m>\beta</m>.
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Write an expected value formula for the strategy of choosing both boxes in terms of <m>\beta</m>.
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Which strategy has a higher expected value?
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</p>
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</statement>
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</task>
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</exercise>
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</page>
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</worksheet> |