Solution: 2022 Winter Final - 17

Author: Michiel Smid

Question

Let $A$ and $B$ be two independent events in some probability space. You are told that $\Pr(A)=1/3$ and that $\Pr(B)=1/2$. What is $\Pr(A\cup B)$?

(a)

$1/3$

(b)

$1/4$

(c)

$1/2$

(d)

$3/4$

(e)

$2/3$

First, let’s calculate $ \Pr(A \cap B) $

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

$ \Pr(A \cap B) = \frac{1}{3} \cdot \frac{1}{2} $

$ \Pr(A \cap B) = \frac{1}{6} $

Now, let’s calculate $ \Pr(A \cup B) $

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

$ \Pr(A \cup B) = \frac{1}{3} + \frac{1}{2} - \frac{1}{6} $

$ \Pr(A \cup B) = \frac{2}{6} + \frac{3}{6} - \frac{1}{6} $

$ \Pr(A \cup B) = \frac{4}{6} $

$ \Pr(A \cup B) = \frac{2}{3} $