TITLE: Difference between a joint probability and the probability of an intersection
QUESTION [21 upvotes]: Is the joint probability $p(X=x,Y=y)$ equivalent to $p(X=x \cap Y=y )$? If it is, why do we use two different notations?

REPLY [18 votes]: Yes, they mean precisely the same thing.
Why different notations? Well, this is not the only place in mathematics where there are multiple notations.  For example, $A'$, $A^c$, and $\bar{A}$ are all used for the complement of $A$. 
The version with the commas is more compact, particularly since the other version should really read $\Pr((X=x)\cap (Y=y))$. Think of the trees saved.
The version  $\Pr((X=x)\cap (Y=y))$ emphasizes the logical structure, so has some pedagogical advantages.