What are some uses for Monte Carlo simulations in mathematics? I've recently been interested in Monte Carlo simulations and their uses, unfortunately most of the examples I find are difficult to understand for a beginner. What are some simple examples of using Monte Carlo simulations used to solve math problems? Bonus points for showing a simple example and using it to understand a more complex one.

Monte Carlo methods are very useful in numerically evaluating high-dimensional integrals. With traditional integration methods, the number of integrand evaluations required to maintain accuracy grows quickly as dimension increases. With Monte Carlo integration, the number of integrand evaluations needed is _independent_ of dimension. For many high-dimensional integrals, Monte Carlo methods are the only practical choice.