Suppose that the particle is absorbed whenever it hits either N or N + 1. Find the probability N (a) that it is absorbed at

Suppose that the particle is absorbed whenever it hits either N or N + 1. Find the probability

πN

(a) that it is absorbed at 0 rather than at N or N + 1, having started at

a, where 0 ≤ a ≤ N + 1. Deduce that, as N → ∞,

πN (a)→

(

1 if p ≤ 1 3 ,

θa if p > 1 3 , where θ = 1 2 {√1 + (4q/p) − 1}.