Solution: 2014 Winter Midterm - 14
Author: Michiel SmidQuestion
A standard deck of 52 cards has 4 Kings. Consider a hand of 9 cards, chosen uniformly at random.
What is the probability that there are exactly two Kings in this hand?
(a)
$\left. \bigl\{ {4 \choose 2} + {48 \choose 7} \bigr\} \middle/ {52 \choose 9} \right.$
(b)
$\left. {4 \choose 2}{48 \choose 7} \middle/ {52 \choose 9} \right.$
(c)
$\left. {52 \choose 9} \middle/ \bigl\{ {4 \choose 2}{48 \choose 7} \bigr\} \right.$
(d)
$1 - \left. {48 \choose 7} \middle/ {52 \choose 9} \right.$
Solution
$S$ = You pick any 9 cards from a deck of 52 = $\binom{52}{9}$
$A$ = You pick 2 Kings from 4 and 7 non-Kings from 48 = $\binom{4}{2} \cdot \binom{48}{7}$
$P(A) = \frac{\binom{4}{2} \cdot \binom{48}{7}}{\binom{52}{9}}$