Are these 2 events independent?

Question

We draw 4 cards from a standard 52-card deck.

$A$ is the event that we draw 4 different color cards.

$B$ is the event that we draw at most 3 aces.

I have calculated $P(A)$ and $P(B)$, and I know that they are independent if $P(A|B)P(B)=P(A)P(B)$, but I don't know how to calculate $P(A|B)$.

Thanks in advance!

Answer 1

These events are not independent. Here is a trick that allows us to see that without computing anything.

Let $\bar{A}$ and $\bar{B}$ denote the complement of events $A$ and $B$, respectively. It is not too hard to see that $A$ and $B$ are independent if and only if $\bar{A}$ and $\bar{B}$ are independent. (Prove it!)

But $\bar{A}$ is the event that we draw 3 colors or less, and $\bar{B}$ is the event that we draw 4 aces. Clearly $P( \bar{A} \cap \bar{B}) = 0$, where the probability of each individual event is positive.