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Problem of the Week 1082

A Fair Game with an Unfair Coin

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Alice: Let's take turns flipping a coin: the first person who gets HEADS wins. I'll go first.
Bob: But you might win on the very first throw; let's compensate by giving me one extra throw on my first turn.
Alice: Fine.
Bob: And let's use my coin, which comes up HEADS with probability p.
Alice: Well, then I better know p before I agree.

What value of p makes the game fair to each player?

Source: Problem posed by Stephen Herschkorn in Math Magazine (June 2007, #1746) for the two cases 1 HEAD wins and 2 HEADS win, with Bob getting an extra toss only if he throws TAILS on his first throw. There are algebraic answers in the published cases and one wonders whether there are algebraic answers in many of the other cases.

© Copyright 2007 Stan Wagon. Reproduced with permission.

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11 October 2007