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