probability priority of comma and bar

Question

In probability does , have priority over |?

I am asking because of below question.

Is below X and (Y given by Z)?

P(X,Y|Z)

or is it (X and Y) given by Z?

Answer 1

The sign | denote the conditional probability. It is not an operation on the events. This sign does not refer to events, but to probability. And if $X,Y,Z$ are some events, then $P(X,Y \mid Z)$ is the conditional probability of the event $X\cap Y$, conditioning on the event $Z$.

Answer 2

To answer your question more directly, yes, comma (,) has higher priority than bar (|) when interpreting this notation.

So P(X,Y|Z) is the conditional probability of X and Y occurring together, given Z. It might help to think of this case as P((X,Y)|Z) (though such use of additional brackets is not common in the notation, afaik).

Similarly, P(X|Y,Z) is the conditional probability of X, given Y and Z (i.e. as if there were additional brackets as follows: P(X|(Y,Z)).