Solution: 2017 Winter Final - 16

Author: Michiel Smid

Question

Let $A$ and $B$ be two events in some sample space. You are given that $$ \begin{align} \Pr(A|B) &= \Pr(B|A), \\ \Pr(A \cup B) &= 1, \\ \Pr(A \cap B) &> 0. \end{align} $$ Which of the following is true?

(a)

$\Pr(A) < 1$

(b)

$\Pr(A) > 1/2$

(c)

$\Pr(A) < 1/2$

(d)

$\Pr(A) > 0$

Solution

Let’s make some statements and see what we can get

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

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

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

$ Pr(A) = Pr(B) $

$ Pr(A \cup B) = Pr(A) + Pr(B) - Pr(A \cap B) = 1 $

$ 2Pr(A) - Pr(A \cap B) = 1 $

For the above statement to be true, $ Pr(A) > \frac{1}{2} $ because $ Pr(A \cap B) > 0 $

To illustrate, let’s use some values

$ Pr(A) = 0.5 $ and $ Pr(A \cap B) = 0.25 $

$ 2 \cdot 0.5 - 0.25 = 1 $

$ 1 - 0.25 = 1 $

$ 0.75 = 1 $

As can be seen, $ Pr(A) > \frac{1}{2} $ needs to be true in order to counter the fact that $ Pr(A \cap B) > 0 $