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.