Suppose in the gambler’s ruin problem that the probability of winning a bet depends on the...

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.

Expert Solution

orchestra answered 2 years ago

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