"Two Cards are picked from a deck with replacement. Let X= number of aces, and Y= number of kings.
X and Y are both discrete random variables that can take on 0,1 and 2."
I'm trying to show whether or not X an Y are independent in the case that the sampling is done with replacement. I first assumed that X and Y are independent simply because replacement was involved but my calculations showed otherwise.
edit: Confirmed in the comments that X and Y are not independent. Can someone please provide a proof that X and Y are dependent and explain the error in the following attempt a proof that they are independent:
$P(X,Y) = P(X \cap Y) = P(X|Y)\times P(Y) $
but $P(X|Y) = P(X)$ because the sampling is done with replacement
so $P(X,Y) = P(X)\times P(Y)$ thus X and Y are independent.
In particular, can anyone explain why this particular scenario of drawing two cards with the random variables as they are defined makes X and Y dependent even though they are sampled with replacement? I understand now that how many X are selected changes the probability of how many Y are selected and I can sense that this has to do with the constraint of how many cards are selected in the event, i.e. the more of one type that are selected changes not the probability of selecting the second type but instead, it changes the amount of the second type that can be selected.
And since the random variable is defined as the amount of the card selected, its clear that this is the heart of the issue. However, I'm having trouble articulating or reasoning further about why this makes sampling with replacement or without replacement irrelevant to the question of dependence/independence of the variables. If anyone could shed light from this angle, I would greatly appreciate it.
Reference: another post asks about the joint probability function of this situation without replacement.