Solution: 2019 Winter Final - 12

Author: Michiel Smid

Question

In a standard deck of 52 cards, each card has a suit and a rank. There are four suits (spades ♠, hearts ♡, clubs ♣, and diamonds ♢), and 13 ranks (Ace, 2,3,4,5,6,7,8,9,10, Jack, Queen, and King).
Assume you get a uniformly random hand consisting of 5 cards. What is the probability that the 5 cards in this hand consist of 3 Aces and 2 Queens?

(a)

$\left. 27 \middle/ {52 \choose 5} \right.$

(b)

$\left. 25 \middle/ {52 \choose 5} \right.$

(c)

$\left. 26 \middle/ {52 \choose 5} \right.$

(d)

$\left. 24 \middle/ {52 \choose 5} \right.$

Solution

Find S

Let S be the sample space of all possible hands of 5 cards.

$|S| = \binom{52}{5}$
Find A

Let A be the event that the hand consists of 3 Aces and 2 Queens.

We choose 3 of the 4 Aces: $\binom{4}{3} = 4$

We choose 2 of the 4 Queens: $\binom{4}{2} = 6$

$|A| = 4 \cdot 6= 24$
Find $ \Pr(A) $

$ \Pr(A) = \frac{|A|}{|S|} $

$ \Pr(A) = \frac{24}{\binom{52}{5}} $