# Expected number of games in a series that terminates

by Tazz01
Tags: expected, games, number, series, terminates
 P: 9 The Question: 2 people A, B play a series of independent games. We have the following probabilities: P(A wins a game) = p P(B wins a game) = q = 1 - p Both players begin with X units of money, and in each game the winner takes 1 unit from the other player. The series terminates when either A or B loses all their money. The assumption is that p > q. Derive the expected number of games in a series. Attempt at a solution: If: Z = number of games, then we are after E[Z]: So if the number of games is X and the series terminated, that means that either that A won all the games from the start or B won all the games from the start. e.g. For E[Z]=N p$^{X}$q$^{0}$ or p$^{0}$q$^{X}$ In order for the series to terminate, the number of wins either for A or B has to be X greater than the number of wins for the other player. e.g. If B wins 1 game, then A needs to win X+1 games in order for the series to terminate. I'm not sure whether my attempt actually helps in obtaining a solution, can anyone advise? Thanks.