# Difference between a joint probability and the probability of an intersection

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?

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.
• Thanks. What is the meaning of $(X=x)\cap(Y=y)$ when X is (say) a number of coin flips and Y is the color of the sky? I don't understand the meaning of an intersection of two events defined on different sample spaces. – usual me Oct 1 '13 at 2:56
• The sample spaces are not different. Let us say that the colour of the sky can be B(lue) or G(ray). Then the sample space is the collection of ordered pairs $(n,C)$ where $n$ is a number and $C$ is one of B or G. – André Nicolas Oct 1 '13 at 3:21
• Let S be the collection of ordered pairs $(n,C)$. My understanding is that $(X=x)$ is an event over S, and is equivalent to $\{(x,B),(x,G)\}$. Is this correct? – usual me Oct 1 '13 at 6:52
• Yes, that's right. Formally, we have to go to the "product space," and that is how the informal event $X=x$ is represented. – André Nicolas Oct 1 '13 at 7:02