Consider the gamblers ruin problem where on each bet the gambler either wins 1 with probability p or loses 1 with probability 1 p. The

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Consider the gambler’s ruin problem where on each bet the gambler either wins 1 with probability p or loses 1 with probability 1− p. The gambler will continue to play until his winnings are either N −i or −i. (That is, starting with i the gambler will quit when his fortune reaches either N or 0.) Let T denote the number of bets made before the gambler stops. Use Wald’s equation, along with the known probability that the gambler’s final winnings are N − i, to find E[T ].

Hint: Let X j be the gambler’s winnings on bet j, j 1. What are the possible values of T j=1 X j ? What is E

,T j=1 X j

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