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1 Gambler’s Ruin Problem

1 Gambler s Ruin ProblemConsider a gambler who starts with an initial fortune of $1 and then on each successive gambleeither wins $1 or loses $1 independent of the past with probabilitiespandq= 1 the total fortune after thenthgamble. The gambler s objective is to reach a totalfortune of $N, without first gettingruined(running out of money). If the gambler succeeds,then the gambler is said towinthe game. In any case, the gambler stops playing after winningor getting ruined, whichever happens first. There is nothing special about starting with $1,more generally the gambler starts with $iwhere 0< i < the game proceeds,{Rn:n 0}forms a simple random walkRn= 1+ + n, R0=i,where{ n}forms an sequence of distributed asP( = 1) =p, P( = 1) =q=1 p, and represents the earnings on the succesive the game stops when eitherRn= 0 orRn=N, let i= min{n 0 :Rn {0, N}|R0=i},denote the time at which the game stops whenR0=i.

1 Gambler’s Ruin Problem Consider a gambler who starts with an initial fortune of $1 and then on each successive gamble either wins $1 or loses $1 independent of the past with probabilities p and q = 1−p respectively.

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