1. Suppose in the gambler's ruin problem that the probability of winning a bet depends on the gambler's present fortune. Speci?cally, suppose that αi is the probability that the gambler wins a bet when his or her fortune is i. Given that the gambler's initial fortune is i, let P(i) denote the probability that the gambler's fortune reaches N before 0.
(a) Derive a formula that relates P(i) to P(i - 1) and P(i + 1).
(b) Using the same approach as in the gambler's ruin problem, solve the equation of part (a) for P(i).
(c) Suppose that i balls are initially in urn 1 and N - i are in urn 2, and suppose that at each stage one of the N balls is randomly chosen, taken from whichever urn it is in, and placed in the other urn. Find the probability that the ?rst urn becomes empty before the second.