Probabilities can be thought of as measures of uncertainty in the occurrence of an event, the truth of a hypothesis, and so on. These measures are numbers between 0 and 1. 0 means the event is impossible to occur and 1 means the event is certain to occur. If you have many events of interest, you can measure their probabilities separately, but you can also measure probabilities of different combinations of these events.

Say you’re following the national soccer championships of England, Spain, and Italy. You want to calculate the probabilities of Arsenal, Barcelona, and Juventus becoming national champions next season. These probabilities are:

- P(Arsenal)
- P(Barcelona)
- P(Juventus)

But what if you want to calculate the probability of *both* Arsenal and Barcelona becoming champions? Or the probability that *at least* one of the three teams does?

In this post, I’m going to show how probabilities of such combinations of events are calculated. I’m going to give the general formulas, as well as the intuition behind them. To do that, I’m first going to introduce a few relevant concepts from probability theory.