Probability of Ruin in the Gambler's Ruin Problem

Topics: Gambler’s Ruin Problem

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Say we are interested in knowing the probability of player A getting ruined in the gambler’s ruin problem.

(observation)

Let $\gamma$ be the first moment upon which the previously defined process reaches one of its two absorbing states ($0$ and $N$). That is:

$$\gamma =\min \left\{n\ge 0\mid X_n=0\vee X_n=N\right\}$$

Then, the probability of player A getting ruined is given by:

$$U_K=\mathbb{P}(X_{\gamma }=0\mid X_0=K)$$

(theorem)

More specifically, we can deduce that:

$$U_K=\{\begin{array}{ll}\frac{N-K}{K}& \text{if }p=q=\frac{1}{2}\\ \frac{{\left(\frac{q}{p}\right)}^K-{\left(\frac{q}{p}\right)}^N}{1-{\left(\frac{q}{p}\right)}^N}& \text{if }p\ne q\ne \frac{1}{2}\end{array}$$