Suppose in the gambler’s ruin problem that the probability of winning a bet depends on the gambler’s present fortune. Specifically, 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 first urn becomes empty before the second.