Gambler’s ruin

Gambler’s ruin is a probability problem in which a gambler’s fortune changes through repeated bets until it reaches zero or a target fortune. It models this process as a random walk with absorbing boundaries.

Random walk: A stochastic process that moves through states according to probabilistic steps. The gambler’s fortune is modeled as a walk whose steps represent wins and losses.
Binomial distribution: The distribution of the number of successes in a fixed number of independent trials with the same success probability. After a fixed number of bets, wins and losses determine the fortune’s displacement.
Risk of ruin: The probability that a person, organization, or system exhausts its resources and cannot continue. The model gives an exact risk of ruin for repeated bets under specified assumptions.
Kelly betting: A wagering strategy that stakes a calculated fraction of current wealth on favorable opportunities. Proportional stakes make wealth multiplicative, unlike the model’s usual fixed-unit steps.
Absorbing state: A state in a stochastic process that, once entered, cannot be left. Ruin and the target fortune end the game because both boundaries absorb the walk.
Gambler’s ruin problem for a fair game: The gambler’s ruin model when each bet has equal probabilities of winning and losing one unit. Its simple ruin probability exposes how starting fortune and target size govern the outcome.
Bankroll management: The practice of allocating and protecting funds committed to uncertain activities. Initial fortune and bet size determine whether a gambler can survive a losing sequence.
Random walk with reflection: A random walk whose boundary redirects or constrains movement instead of absorbing it. A gambler who cannot lose below zero may face a reflecting boundary rather than ruin.
Hitting time: The first time a stochastic process reaches a specified state or set of states. The duration of play is the time required to hit either boundary.
First-step analysis: A method that derives probabilities or expectations by conditioning on the process’s first transition. Conditioning on the first bet yields recurrence equations for ruin probabilities.

Linked from 7 pages