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A gambler’s ruin simulation models a fortune that rises or falls by a fixed amount until it reaches either zero or a target. It estimates how often the player reaches the target, how often they go broke, and how long the process takes. The result depends on the starting fortune, target, step probabilities, and stopping rule—not on a single sample path.
What a gambler’s ruin simulation models
In the finite classical model, the gambler’s fortune is an integer state from 0 through a target of N. Starting at an interior state i, each independent step adds one with probability p or subtracts one with probability q = 1 − p. The process stops at the first visit to 0 or N; both endpoints are absorbing. The University of Cambridge probability notes describe the fortune as a simple random walk on states {0, 1, …, N} with absorbing barriers at 0 and N (Cambridge probability lecture notes).
This is a specific mathematical model, not a forecast of real-world gambling. Changing the stake per step, the probabilities, or what counts as the endpoint changes the model and its results.
What is the probability of gambler’s ruin?
For a fair walk, where p = q = 1/2, the probability of reaching the target N before zero from starting fortune i is i/N. The complementary probability of ruin is 1 − i/N. In the fair two-player version, where players transfer one unit after each fair coin toss, player A starts with a, player B with b, and the combined total is N = a + b. A’s chance of winning is a/(a+b), as described in John Burkardt’s simulation documentation (Burkardt’s gambler’s ruin simulation).
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For a biased walk, the fair-game ratio no longer applies. The probability of an upward step is part of the model, so use a calculation that matches the chosen p, boundaries, and starting state. University probability materials treat these as absorbing random walks; the University of Washington lecture provides an analytic treatment (University of Washington lecture).
How to simulate gambler’s ruin
- Choose the starting fortune i, target N, upward-step probability p, and number of independent trials. State the stake change per step and confirm that the boundaries are 0 and N.
- For each trial, initialize the fortune at i. Generate independent steps: add one with probability p, otherwise subtract one.
- Stop that trial immediately when the fortune reaches 0 or N. Record the endpoint and the number of steps taken.
- Across trials, calculate the fraction that reached N, the fraction that reached 0, and a useful summary of stopping times.
- Compare the simulated outcome rate with an analytic probability or an absorbing-chain calculation using the same assumptions. A finite simulation will differ from the exact result because of sampling variation.
John Burkardt’s page provides implementations in Python, MATLAB, and Octave for the fair two-player case, including repeated games, outcome and step-count summaries, histograms, and a sample trajectory (simulation code and explanation). For R, the CRAN package documentation describes a target-or-bankruptcy game with a specified round-win probability (CRAN ruin package documentation).
How many games does it take to reach bankruptcy?
There is no single answer: the stopping time varies from trial to trial and depends on the starting fortune, target, and step probabilities. In the fair, unit-transfer, two-player setup with initial stakes a and b, Burkardt’s page gives the expected number of coin tosses to finish the game as a × b. That expectation applies to this specific model; it should not be generalized to biased steps or other boundary and stake rules.
When presenting simulation results, show more than an average when possible. A few unusually long runs can pull the mean upward, so a histogram or quantiles can make the distribution of completion times easier to understand. Show example trajectories to illustrate how paths unfold, but do not treat one path as evidence of the overall ruin probability.
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How to read and validate the results
- Keep the model visible: report the initial fortune, target, step size, upward probability, and stopping rule alongside the estimate.
- Separate probability from duration: the chance of reaching a boundary and the number of steps before absorption answer different questions.
- Use a matching benchmark: compare a fair simulation with the fair analytic result; for a biased walk, use an analytic or absorbing-chain result for those same parameters.
- Interpret discrepancies appropriately: repeated-trial estimates fluctuate around the underlying model probability. A finite-run mismatch alone does not show that the model is incorrect.
First-step analysis and Markov-chain absorption are standard analytic approaches to the probability question; an Arizona-hosted preview of An Introduction to Stochastic Modeling discusses deriving gambler’s ruin probability this way (University of Arizona-hosted book preview).
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