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1 Gambler’s Ruin Problem - Columbia

Copyrightc 2009 by Karl Sigman1 Gambler s ruin ProblemLetN 2 be an integer and let 1 i N 1. Consider a gambler who starts with aninitial fortune of $iand then on each successive gamble either wins $1 or loses $1 independentof the past with probabilitiespandq= 1 prespectively. LetXndenote the total fortuneafter thenthgamble. The gambler s objective is to reach a total fortune of $N, without firstgettingruined(running out of money). If the gambler succeeds, then the gambler is said towinthe game. In any case, the gambler stops playing after winning or getting ruined, whicheverhappens first.{Xn}yields a Markov chain (MC) on the state spaceS={0,1,..,N}. The transitionprobabilities are given byPi,i+1=p, Pi,i 1=q,0< i < N, and both 0 andNare absorbingstates,P00=PNN= example, whenN= 4 the transition matrix is given byP= 1 0 0 0 0q0p0 00q0p00 0q0p0 0 0 0 1.

What is the probability that Ellen’s stock reaches the high value of $15 before the low value of $5? SOLUTION We want \the probability that the stock goes up by 5 before going down by 5." This is equivalent to starting the random walk at 0 with a= 5 and b= …

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