Solution: 2019 Winter Final - 18

Author: Michiel Smid

Question

Let $A$ and $B$ be two independent events in some sample space $S$. Recall that $\overline{B}$ denotes the complement of $B$. You are given that $\Pr(A) = 1/4$ and $\Pr(\overline{B}) = 2/3$. What is $\Pr(A \cup B)$?

(a)

3/4

(b)

1/2

(c)

2/3

(d)

1/3

Solution

Let’s just try to find out as much as possible

Find $\Pr(B)$

$\Pr(B) = 1 - \Pr(\overline{B}) = 1 - \frac{2}{3} = \frac{1}{3}$
Determine $A \cap B$

Since $A$ and $B$ are independent, $\Pr(A \cap B) = \Pr(A) \cdot \Pr(B)$

$\Pr(A \cap B) = \frac{1}{4} \cdot \frac{1}{3} = \frac{1}{12}$
Determine $\Pr(A \cup B)$

$\Pr(A \cup B) = \Pr(A) + \Pr(B) - \Pr(A \cap B)$

$\Pr(A \cup B) = \frac{1}{4} + \frac{1}{3} - \frac{1}{12}$

$\Pr(A \cup B) = \frac{3}{12} + \frac{4}{12} - \frac{1}{12}$

$\Pr(A \cup B) = \frac{6}{12} $

$\Pr(A \cup B) = \frac{1}{2} $