Expected Amount of Bets Before Ruin in the Gambler's Ruin Problem

Topics: Gambler’s Ruin Problem

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Say we are interested in knowing the expected amount of bets before player A will get ruined in the gambler’s ruin problem.

(theorem)

This is given by:

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