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Gambler's Ruin and Hitting Times

You start with ii units and bet one unit at a time. Each bet wins with probability pp and loses with probability q=1−pq = 1 - p. You stop when you reach NN units or lose everything. Two questions follow: how likely is each ending, and how long does the game last?

This is gambler's ruin, a random walk between two absorbing barriers, and it is the most reused setup in interview probability. The same equations answer a particle on a line, a best-of series between two players, a price that must hit a stop or a target first, and a trader with limited capital and a small edge.

The first-step equation

Let PiP_i be the probability of reaching NN from ii. Condition on the first bet:

Pi=p Pi+1+q Pi−1,P0=0,PN=1P_i = p\,P_{i+1} + q\,P_{i-1}, \qquad P_0 = 0, \quad P_N = 1

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