The probability of getting a head on a single toss of a coin is p. Suppose that A starts and continues to flip the coin until a tail shows up, at which point B starts flipping. Then B continues to flip until a tail comes up, at which point A takes over, and so on. Let Pn,m denote the probability that A accumulates a total of n heads before B accumulates m. Show that
Pn,m = pPn−1,m + (1 − p)(1 − Pm,n)