If A and B be two events such that $P(A) = \frac{3}{8}$, $P(B) = \frac{5}{8}$ and $P(A \cup B) = \frac{3}{4}$, then $P(A/B) \cdot P\left( {{A^\prime }/B} \right)$ is equal to

Here, $P(A) = \frac{3}{8},P(B) = \frac{5}{8}$ and $P(A \cup B) = \frac{3}{4}$

$\Rightarrow$ $P(A \cap B) = \frac{3}{8} + \frac{5}{8} - \frac{3}{4} = \frac{{3 + 5 - 6}}{8} = \frac{2}{8} = \frac{1}{4}$

and $P\left( {{A^\prime }/B} \right) = \frac{{P\left( {{A^\prime } \cap B} \right)}}{{P(B)}} = \frac{{P(B) - P(A \cap B)}}{{P(B)}}$

$= \frac{{\frac{5}{8} - \frac{1}{4}}}{{\frac{5}{8}}} = \frac{{\frac{{5 - 2}}{8}}}{{\frac{5}{8}}} = \frac{3}{5}$
$\therefore$ $P(A/B) \cdot P\left( {{A^\prime }/B} \right) = \frac{2}{5} \cdot \frac{3}{5} = \frac{6}{{25}}$