Explanation

(d)
Given, $P\left( \frac{B}{A} \right) = \frac{1}{2} \Rightarrow P(B \cap A) = \frac{1}{8}$
and $P\left( \frac{A}{B} \right) = \frac{1}{4} \Rightarrow P(B) = \frac{1}{2}$
$\therefore P(A \cap B) = \frac{1}{8} = P(A).P(B)$
$\therefore$ Events are independent
Now, $P\left( \frac{A^{'}}{B} \right) = \frac{P(A^{'} \cap B)}{P(B)} = \frac{P(B) - P(A \cap B)}{P(B)} = \frac{3}{4}$
and $P\left( \frac{B^{'}}{A} \right) = \frac{P(A \cap B^{'})}{P(A)} = \frac{P(A) - P(A \cap B)}{P(A)} = \frac{1}{2}$