Solution: 2014 Fall Final - 15

Author: Michiel Smid

Question

A bowl contains 5 blue balls and 4 red balls. We choose 2 balls uniformly at random. Define the events

What is the conditional probability $\Pr(A|B)$?

(a)

$\frac{{4 \choose 2}}{{5 \choose 2}{4 \choose 2}}$

(b)

$\frac{{4 \choose 2}}{{9 \choose 2}}$

(c)

$\frac{{5 \choose 2} + {4 \choose 2}}{{4 \choose 2}}$

(d)

$\frac{{4 \choose 2}}{{5 \choose 2} + {4 \choose 2}}$

Let A = Event that both balls are red
A occurs when we choose 2 of the 4 red balls
$ |A| = \binom{4}{2} $
Let B = Event that both balls have the same color
B occurs when we choose 2 of the 5 blue balls or 2 of the 4 red balls
$ |B| = \binom{5}{2} + \binom{4}{2} $
Let $A \cap B$ = Event that both balls are red and both balls have the same color
$ |A \cap B| = \binom{4}{2} $

$ Pr(A|B) = \frac{|A \cap B|}{|B|} $

$ Pr(A|B) = \frac{\binom{4}{2}}{\binom{5}{2} + \binom{4}{2}} $