58. 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 if p i+ if p = 21 Hint: The probability we want is P{Xn+1=i+1|X=i, lim Xm = N} m P(X+1 +1, limm Xm = NIX = i} P{limm Xm NX, = i} =