gambler’s ruin

mathematical problem

Written by

The Information Architects of Encyclopaedia Britannica

Last updated

June 29, 2026

Quick Summary

The gambler's ruin is a mathematical problem that explores the probability of a gambler going broke. Imagine two players, Peter and Paul, starting with a certain amount of money. They play a game where, in each round, one player pays the other a dollar based on a random outcome (like a coin toss). The game continues until one player loses all their money, meaning they are "ruined".1

The problem uses conditional probability to calculate the likelihood of Peter's ruin, considering his initial fortune and the probability of winning each round. This concept is also related to insurance risk theory and queueing theory, where it helps model situations involving potential depletion of resources. Generalizations of this problem are applied in statistical sequential analysis and quality control.21

Summary created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

3 Britannica Sources

Ask Britannica AI about this topic

Britannica doesn't have a full article on this topic, so we've curated a list of our relevant coverage.

probability theory
Applications of conditional probability
…find the probability of “gambler’s ruin.” Suppose two players, often called Peter and Paul, initially have x and m − x dollars, respectively. A ball, which is red with probability p and black with probability q = 1 − p, is drawn from an urn. If a red ball…
Read More
…find the probability of “gambler’s ruin.” Suppose two players, often called Peter and Paul, initially have x and m − x dollars, respectively. A ball, which is red with probability p and black with probability q = 1 − p, is drawn from an urn. If a red ball…
Read More