Three events A, B and C have probabilities $\frac{2}{5},\frac{1}{3}$ and $\frac{1}{2}$, respectively. If, $P(A \cap C) = \frac{1}{5}$ and $P(B \cap C) = \frac{1}{4}$, then find the values of $P(C/B)$ and $P\left( {{A^\prime } \cap {C^\prime }} \right)$.

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

$\therefore$ $P(C/B) = \frac{{P(B \cap C)}}{{P(B)}} = \frac{{1/4}}{{1/3}} = \frac{3}{4}$

and $P\left( {{A^\prime } \cap {C^\prime }} \right) = 1 - P(A \cup C) = 1 - [P(A) + P(C) - P(A \cap C)]$

$= 1 - \left[ {\frac{2}{5} + \frac{1}{2} - \frac{1}{5}} \right] = 1 - \left[ {\frac{{4 + 5 - 2}}{{10}}} \right] = 1 - \frac{7}{{10}} = \frac{3}{{10}}$