Probability and Set Theory

Question

When working with probabilities what does P(X,Y) means in means of set theory? Is it $P(X\cap Y)$ or $P(X\cup Y)$ ?

I'm confused on this basic subject.

Mainly I'm searching for P(X,Y) where X=(U,V) and Y=(V,W), also U,V and W are independent.

Answer 1

As far as I'm aware there's no universal connotation to $P(X,Y)$. It could mean one of the two things you mentioned.

Answer 2

If $A$ and $B$ are events then I would regard $P(A,B)$ as meaning $P(A \cap B)$

i.e. the probability both events occur

In the expression for mutual information $$I(X;Y) = \sum_{y \in Y} \sum_{x \in X} p(x,y) \log{ \left(\frac{p(x,y)}{p(x)\,p(y)} \right) }$$ that is the interpretation, since the events are $X=x$ and $Y=y$.

If $X$ and $Y$ are independent then $P(X=x \cap Y=y)= P(X=x)P(Y=y)$ so $\log \left(\frac{p(x,y)}{p(x)\,p(y)} \right)=0$ and the mutual information is zero as you might expect