Monte Carlo simulation

Monte Carlo simulation estimates a quantity by averaging outcomes generated from random or pseudorandom inputs. A model first specifies uncertain variables and their probability distributions; the algorithm then samples those variables, evaluates the model, and aggregates the results over many trials.¹ The output may be a mean, probability, quantile, expected cost, or entire empirical distribution rather than a single deterministic answer.

The method is especially useful for propagation of uncertainty, integration, optimization, and stochastic processes. Its statistical foundation is the law of large numbers: under suitable conditions, the sample average approaches the desired expectation as the number of independent trials increases. The central-limit theorem often supplies an approximate standard error, allowing a simulation to report a confidence interval alongside its estimate.²

Monte Carlo accuracy improves slowly but predictably: reducing random sampling error by half generally requires about four times as many trials. This inverse-square-root behavior makes the method easy to scale but potentially expensive for rare events, highly variable outputs, or computationally costly models.²

Good practice includes checking convergence, repeating runs with independent random-number streams, examining the effect of sample size, and reporting uncertainty rather than only a point estimate. Quasi-Monte Carlo replaces ordinary random points with low-discrepancy sequences, which can improve integration for smooth, low-dimensional problems. Variance-reduction methods—including importance sampling, stratified sampling, antithetic variates, and control variates—change how samples are chosen or combined to obtain more information per model evaluation.

Monte Carlo simulation supports decisions in fields where uncertainty and complex interactions are central. Applications include financial risk, nuclear- and particle-transport calculations, reliability engineering, inventory and queueing models, climate and environmental assessment, Bayesian inference, and the pricing of financial derivatives. In these settings, the simulated object may be a physical trajectory, a sequence of failures, a market scenario, or a draw from a posterior distribution.

The modern method was developed during the 1940s by researchers associated with the Manhattan Project, notably Stanislaw Ulam, John von Neumann, Nicholas Metropolis, and colleagues. The name “Monte Carlo” alluded to the gambling center in Monaco and became established through this wartime work. Metropolis and Ulam’s 1949 paper helped define the method as a general numerical technique rather than a tool limited to one physical problem.

Monte Carlo simulation can model rare events, but ordinary sampling may almost never encounter the event of interest. Reliability and safety studies therefore use specialized schemes such as importance sampling, splitting, and subset simulation; these methods concentrate computation in influential regions while preserving a statistically interpretable estimate.

Randomness in a computer is normally produced by a deterministic pseudorandom-number generator, so reproducibility depends on recording the generator, seed, streams, and sampling procedure. Parallel simulations require care because naïvely duplicated streams can create correlations. Another subtle issue is model uncertainty: many trials reduce numerical sampling error, but they cannot correct a biased model, an inappropriate distribution, or dependence that the model omitted.

In Bayesian computation, Monte Carlo methods may draw from distributions that cannot be integrated analytically. Markov chain Monte Carlo uses a dependent sequence whose long-run distribution is the target; unlike simple independent sampling, it requires diagnostics for mixing, autocorrelation, and effective sample size.³

Pseudorandom-number generator

A deterministic algorithm that produces sequences designed to have statistical properties resembling random numbers.

Variance reduction

A family of techniques that lowers the sampling variance of an estimator without necessarily increasing the number of model evaluations.

Quasi-Monte Carlo

A numerical integration approach using low-discrepancy point sequences rather than independently random samples.

Markov chain Monte Carlo

A class of methods that constructs a Markov chain whose stationary distribution is the probability distribution being studied.