A deck of 52 cards is shuffled and a bridge hand of 13 cards is dealt out. Let X and Y denote, respectively, the number of aces and the number of spades in the hand.
(a) Show that X and Y are uncorrelated.
(b) Are they independent?
I know that if $Cov(X,Y)=0$ then $X$ and $Y$ are uncorrelated. I also know that $X$ and $Y$ are independent since the probability of choosing aces does not affect the probability of drawing spades. I tried using $Cov(X,Y)=E[XY]-E[X]E[Y]$ but got stuck on trying to find the expected value of $X$ and $Y$. Is this a valid approach to the problem or is there something I am missing?
Hint: The random variables $X$ and $Y$ are for sure not independent. Informally, if we have $4$ Aces, then for sure we have a spade. This observation can be turned into a formal proof of non-independence.
Added: Since in a comment it is said that the textbook asserts $X$ and $Y$ are independent, let us show that they are not. Note that $\Pr(X=4)\ne 0$ and $\Pr(Y=0)\ne 0$. But $\Pr( (X=4)\cap (Y=0))=0$, since we cannot have $4$ Aces and no spades.