Solution to Gambler's Ruin

Theorem

Let a gambler $G$ with an initial capital of $C$ units play a sequence of games in which:

$G$ gains $1$ unit of capital with probability $p$

$G$ loses $1$ unit of capital with probability $q = 1 - p$.

The game end either when:

$G$ is ruined if and only if he loses all $C$ units

$G$ wins if and only if he attains a total fortune of $N$ units, where $N > C$.

Then the probability that $G$ is ruined is given by:

$\map \Pr {\text {ruin} } = \begin {cases} \dfrac {N - C} N & : p = \dfrac 1 2 \\ \\ q^C \dfrac {p^{N - C} - q^{N - C} } {p^N - q^N} & : \text {otherwise} \end {cases}$

Proof

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Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): gambler's ruin