So apparently there’s a way to combine Prof James two favorite things: Graphics and really torturous numerical problems. In a full global illumination setting, where indirect lighting (light that bounces of a surface and lights another) is also computed, the goal of a renderer is to solve the rendering equation:

which is computing an integral over the hemisphere at point x of a 6 dimensional function, f_r.
This, obviously, has no closed form integral. In fact, it would be a huge pain to do this using a Riemann sum. So, people use Monte Carlo techniques to estimate the integral. Basically, at each point, they pick a bunch of random points on the hemisphere, find the values of the integral there, sum up all the values found, and divide by the number of points chosen. To get a reasonable looking image, there are lots of techniques people use. The main technique is called importance sampling, where rather than uniformly randomly selecting a point on the hemisphere, a probability distribution function is used, so hopefully places where more light comes in are picked more often. This decreases the variance of the integral (a good thing!) so fewer samples are needed.

http://en.wikipedia.org/wiki/Rendering_equation