Gambler's Ruin
Consider a gambler who at each play of the game, either wins one dollar with probability
p or lose one dollar with probability (1 - p).
The game is over if either he loses all his money or he attains a fortune of N dollars.
Let the gambler's fortune be the state of the gambling process then the process is a random
walk with absorbing boundaries 0 and N.
We note that the process will eventually stay at state 0 or N for ever if one of the states
is reached. If initially the gambler has a fortune of 0 < i < N, what is the probability
that he will eventually win the money he needs.
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