Probability math question?

Consider a game of chance with the following characteristics: A person repeatedly bets $1 each play. If he wins, he receives $1. If he goes broke, he stops playing. Also, if he accumulates $3, he stops playing. On each play the probability of winning is 0.23 and of losing 0.77. If the gambler begins with $2, what is the expected number of times that he will play before quitting?