A fantastic tree diagram depicting Bayes’ Theorem.

The diagram superimposes trees for two events (A and B) and their respective complements.

Green branches correspond to event A, yellow to event B, red to A complement, and blue to B complement.

Starting from the center (the sample space, omega), we choose an event and follow its color. The probability that it occurs is written by its branch. For example, the probability of event A is P(A), written next to the green branch.

Now that one event is given, we proceed to another by conditional probability. E.g. Given event A, what is the likelihood that B occurs? This is written P(B|A), next to the yellow branch.

The bottom node represents the intersection of the two events, i.e. the chance that *both *occur. Numerically, this is the product of the probability of the first event with that of the conditional event.

Loosely, this is Bayes’ Theorem (written at the bottom). For our example, it means that the probability of both events (A and B), is the probability of A times B given A, *and equivocally,* B times A given B. Diagrammatically, you can reach the bottom node by going down-left then right, or down-right then left.

Mathematics is beautiful. <3