Computing probabilities in a game of bridge

Question

In the game of bridge, a player is given 13 cards from a deck of 52 cards, what is the probability that he/she gets exactly one king and one queen? Furthermore what is the expected value of the number of aces he/she gets?

Answer 1

Hypergeometric distribution

Probability of one king and one queen: $$\frac{\binom{4}{1}\binom{4}{1}\binom{44}{11}}{\binom{52}{13}}=\frac{36556}{189175}$$

Expected number of aces: $$\sum_{k=0}^4 k \frac{\binom{4}{k}\binom{48}{13-k}}{\binom{52}{13}} = 4 \cdot \frac{13}{52}=1$$

Answer 2

Another way to find the expected number $X$ of aces:

Let $X_i$ be the indicator that card $i$ is an ace, for $1\leq i\leq 13$. Then $\mathbb E[X_i]=\mathbb P(X_i=1)=1/13$ (if we imagine taking the top $13$ cards in a randomly shuffled deck, then clearly each card has a $1/13$ chance of being an ace). So by linearity of expectation $$\mathbb E[X]=\mathbb E[X_1+\dots+X_{13}]=\mathbb E[X_1]+\dots+\mathbb E[X_{13}]=13\cdot\frac{1}{13}=1.$$