 Monte Carlo integration

To calculate $I=\int h(x)p(x)dx$, generate independent $X_s$ with density $p$ and average $h(X_s)$. Integrability gives convergence by the strong law of large numbers; finite variance gives an unbiased estimate with variance $\mathrm{Var}(h(X))/S$. Applying this to indicators estimates probabilities. Correlated Markov chain Monte Carlo samples can also approximate the integral, but their error is determined by the chain's dependence rather than the independent-sample variance formula.