In the gambler’s ruin problem of Section 4.5.1, suppose the gambler’s fortune is presently i, and suppose that we know that the gambler’s fortune will eventually reach N (before it goes to 0). Given this information, show that the probability he wins the next gamble is p[1 − (q/p)i+1]

1 − (q/p)i , if p = 1 2

i + 1 2i , if p = 1 2