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.