Explanation

(a)
Given that, $P(A \cap B) = \frac{1}{3},\ P(A \cup B) = \frac{5}{6}$ , and $P(A) = \frac{1}{2}$
$\therefore P(A \cup B) = P(A) + P(B) - \ P(A \cap B)$
$\Rightarrow \ \frac{5}{6} = \frac{1}{2} + P(B) - \frac{1}{3}$
$\Rightarrow P(B) = \frac{5}{6} + \frac{1}{3} - \frac{1}{2} = \frac{4}{6} = \frac{2}{3}$
$\therefore\ P(A) \times P(B) = \frac{1}{2} \times \frac{2}{3} = \frac{1}{3} = P(A \cap B)$
This shows that $A\ and\ B$ are independent events.