Problem of the Week 1128

Let's Flip Some Coins

Alice: I have one fair coin and another that comes up heads 51% of the time. We'll each take a coin and then make N flips with it; if you get more heads than I do, you win.

Bob: I suspect you would like the coin that is more likely to come up heads.

Alice: Right you are.

Bob: Then I should get to choose what N is.

Alice: Go ahead.

What N should Bob choose so as to to maximize his chance of winning?

Extra credit: Can one find a formula, or a good estimate, for the optimal value of N when the two coins have probability p and q, respectively, of coming up heads? For an interesting case, set the weaker probability (Bob, p) to 0.1 and the stronger (Alice, q) to 0.101. I have a conjecture for general p and q and experiments show it to be accurate.

Source: My colleague Victor Addona, Macalester College.

© Copyright 2009 Stan Wagon. Reproduced with permission.