Four cards are selected at random from a well-shuffled deck of $52$ cards. Let the random variable $X$ denote the number of queens drawn and let the random variable $Y$ denote the number of kings drawn. Find $f_{xy}(2,1)$
$f_{xy}(2,1)$ is the probability that exactly two cards are queens and exactly one card is a king.
$f_{xy}(2,1) = \binom{4}{2}(\frac{4}{52})^{2}(1-\frac{4}{52})^2\binom{4}{1}(\frac{4}{52})^{1}(1-\frac{4}{52})^{4-1}$
So, $f_{xy}(2,1) = 0.0073$. This answer is wrong. Accepted answer is $0.0039$ What am I doing wrong?
Best Answer
Apart from other things, your basic mistake is that you are trying to work out some figure for drawing with replacement in a situation where cards are being drawn without replacement.
The correct figure for drawing two queens and one king in four cards
$= \Large\frac{\binom42\binom41\binom{44}1}{\binom{52}4} \approx 0.0039$
Added
Also, in the header, you are asking for the p.m.f (probability mass function) when you actually just want the probability of getting $2$ queens, $1$ king in $4$ cards.