A central probability formula is the following:

Let’s think about why this equation is the case. If we are given that B occurs, the space of all possibilities becomes:

The shaded area in the picture above is the denominator, since we know that no matter what, we are looking at what happens given that B has occurred. To derive Bayes’ rule, we can conceptually think of the “given B” part of *P**A ** B) *as saying that no matter what happens, B has occurred, as shown above. We can therefore put B in the denominator.

We can now write the probability of A given B as *P**A ∩B *over *P(B)*, since the shaded space covering all of B is the denominator, and the probability of A within this space is the part of A that intersects with B, or the probability that both A and B occur *P**A ∩B*. This gives us the following formula for Bayes theorem:

To turn this equation into the Bayes rule as we originally introduced it, we need to use the following probability rule:

This gives us the final Bayes formula: