You are given that $A$ and $B$ are two events such that $P(B) = \frac{3}{5}$, $P(A \mid B) = \frac{1}{2}$ and $P(A \cup B) = \frac{4}{5}$, then $P(B \mid A')$ is equal to

$\frac{3}{5}$

The correct answer is Option (4) → $\frac{3}{5}$ ##

Here, $P(B) = \frac{3}{5}, P(A \mid B) = \frac{1}{2}$ and $P(A \cup B) = \frac{4}{5}$

$∵P(A \mid B) = \frac{P(A \cap B)}{P(B)} \Rightarrow \frac{1}{2} = \frac{P(A \cap B)}{3/5}$

$∴P(A \cap B) = \frac{3}{5} \times \frac{1}{2} = \frac{3}{10}$

and $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

$\Rightarrow \frac{4}{5} = P(A) + \frac{3}{5} - \frac{3}{10}$

$∴P(A) = \frac{4}{5} - \frac{3}{5} + \frac{3}{10} = \frac{8 - 6 + 3}{10} = \frac{5}{10} = \frac{1}{2}$

$P(B \mid A') = \frac{P(B \cap A')}{P(A')} = \frac{P(B) - P(B \cap A)}{1 - P(A)}$

$= \frac{\frac{3}{5} - \frac{3}{10}}{1 - \frac{1}{2}} = \frac{\frac{6 - 3}{10}}{\frac{1}{2}} = \frac{3/10}{1/2} = \frac{3}{5}$