A numerical method that estimates definite integrals by averaging the values of the integrand evaluated at randomly sampled points from a chosen probability distribution over the domain.

Principle

Use random sampling so that the sample mean of integrand evaluations converges to the integral by the law of large numbers; variance controls accuracy.

Demonstration

Estimate the integral of f(x)=exp(-|x|) over a high-dimensional unit cube by drawing N independent uniform samples and computing the average f(x); increase N to reduce Monte Carlo error ≈ O(N^{-1/2}).

Misapplication

Applying naive random sampling in very high dimension without variance reduction (importance sampling, stratification) or using too small N, yielding misleadingly noisy estimates.

Consequence

Provides dimensionally scalable approximation where deterministic quadrature is infeasible; error diminishes stochastically and can be quantified by sample variance.

Reversal

Deterministic quadrature or interpolation methods yield faster convergence for smooth low-dimensional integrands but typically fail to scale to very high dimensions.

Boundary

Requires the integrand to be integrable under the sampling law; not suitable when the integrand has infinite variance under the proposal or when exact symbolic integration is available and preferred.

Semantic Tension

Contrasts with deterministic cubature and with quasi-random (low-discrepancy) sequences; tension is between stochastic O(N^{-1/2}) convergence and faster rates for structured deterministic methods.

Synthesis

Monte Carlo integration approximates integrals by random sampling and averaging; it trades statistical convergence speed for robustness to dimension and complicated domains, with accuracy governed by variance control.