4. Consider a random walk on the integers in which the particle moves either two units to the right (with probability p) or one unit to the left (with probability q = 1 − p) at each stage, where 0 < p < 1. There is an absorbing barrier at 0 and the particle starts at the point a

(> 0). Show that the probability π

(a) that the particle is ultimately absorbed at 0 satisfies the difference equation

π

(a) = pπ(a + 2) + qπ(a − 1) for a = 1, 2, . . . , and deduce that π

(a) = 1 if p ≤ 1