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 PA 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 PA ∩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 PA ∩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: