You have two opponents with whom you alternate play. Whenever you play A, you win with probability pA; whenever you play B, you win with probability pB, where pB >pA. If your objective is to minimize the expected number of games you need to play to win two in a row, should you start with A or with B?

Hint: Let E[Ni ] denote the mean number of games needed if you initially play i. Derive an expression for E[NA] that involves E[NB]; write down the equivalent expression for E[NB] and then subtract.