If A and B are two events such that $P(A) = \frac{1}{2},P(B) = \frac{1}{3}$ and $P(A/B) = \frac{1}{4}$, then $P\left( {{A^\prime } \cap {B^\prime }} \right)$ equals

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

$\Rightarrow$ $P(A \cap B) = P(A/B) \cdot P(B) = \frac{1}{4} \cdot \frac{1}{3} = \frac{1}{{12}}$

Now, $P\left( {{A^\prime } \cap {B^\prime }} \right) = 1 - P(A \cup B)$

$= 1 - [P(A) + P(B) - P(A \cap B)]$
$= 1 - \left[ {\frac{1}{2} + \frac{1}{3} - \frac{1}{{12}}} \right] = 1 - \left[ {\frac{{6 + 4 - 1}}{{12}}} \right]$

$= 1 - \frac{9}{{12}} = \frac{3}{{12}} = \frac{1}{4}$