Lesson 1

In many probability problems, the sample space is easy to describe but hard to enumerate. Sometimes it is enormous; sometimes the experiment is given only by a procedure; sometimes the exact formula is simply not worth the effort. In such situations, it is natural to estimate probabilities by repeating the experiment many times.

This idea is called the Monte Carlo method. To estimate the probability of an event \(A\), we run the experiment \(N\) times and count the fraction of runs in which \(A\) occurs: \[\widehat{\Pr}_{N}[A]=\frac{\text{number of successful runs}}{N}.\] The estimate is random: if we run the whole simulation again, we usually get a different number. But as \(N\) grows, these estimates become more stable.