103.My Twenty Against Their Thirty
★★★expected valueJane Street
You roll a fair 20-sided die numbered 1 to 20 and your opponent rolls a fair 30-sided die numbered 1 to 30. If your number is strictly higher, your opponent pays you the value you rolled. If their number is strictly higher, you pay them the value they rolled. If the two numbers are equal, you pay them that number. What is your expected profit, in dollars, from a single round?
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All 600 pairs are equally likely, so total the amount collected over the outcomes you win and subtract the amount paid over the rest. Notice that a tie sits on the paying side, which lets you merge two of the cases.
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