I came across the following interesting problem today:
A game consists of a sequence of plays; on each play either you or your opponent scores a point, you with probability 𝑝 < 1/2, he with probability (1 - 𝑝). The number of plays is to be even (2, 4, 6, ...). To win the game, you must score more than half the points. You are allowed in advance to choose the number of plays. How many plays should you choose in terms of 𝑝 to optimize your chances of winning?
The problem is much less trivial than I thought at first glance. I know the answer, but will refrain from posting it unless the problem goes unsolved for an extended time.