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Problem of the Week 1123

A Fair Dice Game

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Alice: Look at this neat new 15-sided die. The probability of it landing on any of 1, 2, 3, ..., 15 is exactly 1/15. Let's play a game. We'll take turns rolling it and whoever rolls a 1 first wins. I'll roll first.

Bob: Hold on. If you go first, then you have a better chance of winning. Here's a better idea: We'll pick two positive integers A and B. You can go first, but whenever you roll the die, you have to get A or lower to win, and when I roll I have to get B or lower.

Alice: Okay. But what should A and B be?

What should A and B be in order that Bob's and Alice's chances of winning are equal?

Source: Dan Velleman via the Amherst College Walker Prize Examination.

© Copyright 2009 Stan Wagon. Reproduced with permission.

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15 October 2009