Probabilities of events related to drawing cards

Question

from a deck of 52 cards, 9 cards are drawn at random without replacement. If X and Y are the number of hearts and diamonds respectively. What is the joint probability of X and Y and also what is the probability of X being greater than Y?

Answer 1

Denote the number of hearts as $X$ and the number of diamonds as $Y$. The probability of obtaining $x$ hearts and $y$ diamonds is $P(X=x,Y=y)=\frac{\binom{13}{x}\binom{13}{y}\binom{26}{9-x-y}}{\binom{52}{9}}$, with $x\geq0$, $y\geq0$, and $x+y\leq9$.

The probability that $X$ is greater than $Y$ is a simple sum.

$$P(X>Y)=\sum_{y=0}^{4}\sum_{x=y+1}^{9-y}P(X=x,Y=y)$$