In the gambler’s ruin problem of Section 4.5.1, suppose the gambler’s fortune is presently i, and suppose that we know that the gambler’s fortune will eventually reach N (before it goes to 0). Given this information, show that the probability he wins the next gamble is

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P[1-(q/p)+1] 1- (q/p)i i+1 2i" if p if p = Hint: The probability we want is P{Xn+1=i+1|Xn=i, lim Xm = N} P{Xn+1=i+1, limm Xm = N|Xn=i} P{limm Xm=N|X = i} n