When we reason about the mean of multiple events, not about the probabilities of a single event, we can call on the Central Limit Theorem, which states:

Given a sufficiently large sample:

  • The means of the samples in a set of samples will be approximately normally distributed.
  • This normal distribution will have a mean close to the mean of the population.
  • The variance of the sample means will be close to the variance of the population divided by the sample size.

This means that for an event:

  • We can obtain samples (large enough)
  • We can calculate their means
  • The mean of these means close to population mean
  • The variance of these means close to variance of population / sample size