The rendering equation contains integrals over many possible light directions and recursively over many possible light paths. Path tracing estimates those integrals by random sampling.

For an integral written as an expectation, Monte Carlo estimation uses independent samples $X_k$:

$$ \mathbb{E}[f(X)]\approx \frac1N\sum_{k=1}^N f(X_k). $$

A path tracer sends a camera ray into the scene, samples a new scattering direction at each hit, multiplies the path's throughput by the corresponding material and geometric factors, and accumulates emitted or directly sampled light.

Random sampling makes the estimator noisy for finite $N$. Increasing the number of samples reduces variance statistically rather than making every pixel deterministically converge in a fixed number of steps.

Importance sampling chooses directions more often where they are expected to contribute strongly and corrects for that nonuniform probability, reducing variance without changing the expected result.

Path tracing turns global illumination into repeated stochastic evaluation of light paths rather than enumerating every possible path explicitly.