Solution: 2014 Fall Final - 12

Author: Michiel Smid

Question

A bowl contains 5 blue balls, 4 red balls, and 6 green balls. We choose 2 balls uniformly at random. What is the probability that these 2 balls have the same color?

(a)

$\frac{{15 \choose 2}}{{5 \choose 2}{4 \choose 2}{6 \choose 2}}$

(b)

$\frac{{5 \choose 2} + {4 \choose 2} + {6 \choose 2}}{{15 \choose 2}}$

(c)

$\frac{{5 \choose 2}{4 \choose 2}{6 \choose 2}}{{15 \choose 2}}$

(d)

$\frac{{15 \choose 2}}{{5 \choose 2} + {4 \choose 2} + {6 \choose 2}}$

Solution

Let A = Event that both balls are blue
A occurs when we choose 2 of the 5 blue balls
$ |A| = \binom{5}{2} $
Let B = Event that both balls are red
B occurs when we choose 2 of the 4 red balls
$ |B| = \binom{4}{2} $
Let C = Event that both balls are green
C occurs when we choose 2 of the 6 green balls
$ |C| = \binom{6}{2} $
Let S = All possible outcomes
S occurs when we choose 2 of the 15 balls
$ |S| = \binom{15}{2} $

$ |A \cup B \cup C| = |A| + |B| + |C| $

$ |A \cup B \cup C| = \binom{5}{2} + \binom{4}{2} + \binom{6}{2} $

$ Pr(A \cup B \cup C) = \frac{\binom{5}{2} + \binom{4}{2} + \binom{6}{2}}{\binom{15}{2}} $