Solution: 2014 Winter Final - 17

Author: Michiel Smid

Question

Let $A$ and $B$ be events in a sample space, such that $\Pr(A) = 1/3$, $\Pr(B) = 1/2$, and $\Pr(A|B) = 2/5$. What is $\Pr(B|A)$?

(a)

4/5

(b)

1/5

(c)

3/5

(d)

2/5

Find $ Pr(A \cap B) $:
$ Pr(A|B) = \frac{Pr(B \cap A)}{Pr(B)} $
$ \frac{2}{5} = \frac{Pr(B \cap A)}{ \frac{1}{2}} $
$ Pr(B \cap A) = \frac{2}{5} \times \frac{1}{2} = \frac{1}{5} $
Find $ Pr(B|A) $:
$ Pr(B|A) = \frac{Pr(A \cap B)}{Pr(A)} $
$ Pr(B|A) = \frac{ \frac{1}{5}}{ \frac{1}{3}} $
$ Pr(B|A) = \frac{1}{5} \times \frac{3}{1} $
$ Pr(B|A) = \frac{3}{5} $