Solution to Gambler's Ruin
Jump to navigation
Jump to search
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
This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Sources
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): gambler's ruin