The problem has two parts. (i) Given $P(A) = \frac{1}{2}$ and that events $A$ and $B$ are mutually exclusive, we need to find $P(AB)$. (ii) Given $P(A) = \frac{1}{2}$ and $P(AB) = \frac{1}{8}$, we need to find $P(A \overline{B})$.

1. Problem Description

The problem has two parts.

(i) Given $P(A) = \frac{1}{2}$ and that events $A$ and $B$ are mutually exclusive, we need to find $P(AB)$.

(ii) Given $P(A) = \frac{1}{2}$ and $P(AB) = \frac{1}{8}$, we need to find $P(A \overline{B})$.

2. Solution Steps

(i) If $A$ and $B$ are mutually exclusive events, it means that they cannot occur at the same time. Therefore, the intersection of $A$ and $B$ is an empty set, and the probability of $A$ and $B$ occurring together is

0. $P(AB) = 0$

(ii) We are given $P(A) = \frac{1}{2}$ and $P(AB) = \frac{1}{8}$. We want to find $P(A \overline{B})$.

We know that $A$ can be written as the union of two mutually exclusive events: $AB$ and $A\overline{B}$.

Therefore, $P(A) = P(AB) + P(A\overline{B})$.

$P(A\overline{B}) = P(A) - P(AB)$

$P(A\overline{B}) = \frac{1}{2} - \frac{1}{8}$

$P(A\overline{B}) = \frac{4}{8} - \frac{1}{8}$

$P(A\overline{B}) = \frac{3}{8}$

3. Final Answer

(i) $P(AB) = 0$

(ii) $P(A\overline{B}) = \frac{3}{8}$