I'm currently writing a blog post that uses Bayes' Law but don't want to muddy the post with a review in layman's terms. So I have something to link, here is a short description and a chance to flex my teaching muscles before the school year starts.

Bayes' Law

For those who aren't sure, Bayes' Law tells us that the probability event $X$ occurs given we know that event $Y$ has occurred can easily be computed. It is written as $\text{Pr}(X \, | \, Y)$ vertical bar is meant like the word "given", in other words, the event $X$ is distinct from the event $(X \, | \, Y),$ i.e. $X$ given $Y$. Bayes' law, states that

This effectively is a re-scaling of the events by the total probability of the given event: $\text{Pr}(Y)$.

For example, if $X$ is the event that a $3$ is rolled on a fair die and $Y$ is the event that the roll is odd. We know of course that $\text{Pr}(Y) = \frac{1}{2}$ since half of the rolls are odd. The event $X \text{ and } Y \text{ both occur}$ in this case is the same as $X$ since the roll can only be $3$ if the roll is already odd. Thus

and we can compute the conditional probability

As we expect, one out of every three odd rolls is a $3$.

Bayes' Law Extended Form

Instead of considering a single event $Y,$ we can consider a range of $n$ possible events $Y_1, Y_2, \ldots, Y_n$ occur. We require that one of these $Y$-events must occur and that they cover all possible events that could occur. For example $Y_1$ is the event that H₂O is vapor, $Y_2$ is the event that H₂O is water and $Y_3$ is the event that H₂O is ice.

In such a case we know that since the $Y$-events are distinct

Using Bayes' law, we can reinterpret as

and the above becomes

The same is true if we replace $3$ with an arbitrary number of events $n$.