Continuation. Consider the gambler’s random walk in the preceding exercise in which there is no upper bound m at which the gambler stops playing. That is, the fortune Xn moves in S = {0, 1,...} until...

Continuation. Consider the gambler’s random walk in the preceding exercise in which there is no upper bound m at which the gambler stops playing. That is, the fortune Xn moves in S = {0, 1,...} until it is absorbed at 0 or remains positive forever. Let γ_i
denote the probability of being absorbed at 0 when X₀
= i. Show that γi = 1 if q ≥ p, and that the mean length of the game starting i i/(q − p) if q>p. Thus a gambler is bound to loose in a series of gambles if the odds are not favorable.