---
tag: 0WFMKK7DFZ
kind: problem
name: Gambler's ruin
aliases: [gamblers-ruin-problem]
msc: [60G50, 60G40]
uses: [0C08YMV95H, 0E5G5FMT2R]
sources:
  - url: https://en.wikipedia.org/wiki/Gambler's_ruin
    locator: Fair coin flipping
  - url: https://en.wikipedia.org/wiki/Gambler's_ruin
    locator: Unfair coin flipping
same_as:
  wikidata: Q660446
  wikipedia: Gambler's ruin
---

## Problem

A gambler starts with $k$ units of money, where $0 \le k \le N$ for an integer $N \ge 1$. In each round, the gambler wins one unit with probability $p \in (0, 1)$ and loses one unit with probability $q = 1 - p$. The rounds are independent.
The game stops when the fortune reaches $0$ (ruin) or $N$ (the target).
Find the probability $P_k$ that the gambler reaches $N$ before $0$.

## Remarks

The fortune is a [[random walk|0C08YMV95H]] on $\{0, 1, \dots, N\}$ that stops at $0$ and $N$. By the [[law of total probability|0E5G5FMT2R]] for the first round, $P_k = p P_{k+1} + q P_{k-1}$ for $0 < k < N$, with $P_0 = 0$ and $P_N = 1$.
For a fair game, $p = q = 1/2$, the solution is $P_k = k/N$.
For $p \ne q$, the solution is $P_k = \dfrac{1 - (q/p)^k}{1 - (q/p)^N}$.
In a fair game against an opponent with unlimited money ($N \to \infty$), the gambler is ruined with probability $1$.

## History

An early form of the problem appears in a letter of Blaise Pascal to Pierre de Fermat in 1656.

## Explanations

### L1 human

A gambler who plays a fair game against a much richer opponent will almost surely lose everything in the end.
With $10$ units and a target of $100$ in a fair game, the chance to reach the target is only $10/100 = 0.1$.