3. A particle performs a random walk on the set {−N,−N + 1, . . . , N − 1, N} and is absorbed if it reaches −N or N, where N > 1. The probability of a step of size −1 is q = 1 − p, with 0 < p < 1. Suppose that the particle starts at 0. By conditioning on the first step and using Theorem 10.23, or otherwise, show that when p 6= q, the probability of the particle being absorbed at N or −N before returning to 0 is

(p − q)(pN + qN )

pN − qN

.