Question

Let $A$ 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

• A. $\frac{2}{5}$
• B. $\frac{3}{8}$
• C. $\frac{3}{20}$
• D. $\frac{6}{25}$

Answer 1

Answer: (D)

$P(A)=\frac{3}{8} \quad P(B)=\frac{5}{8} \quad P(A \cup B)=\frac{3}{4}$
${c}P(A \cap B)=P(A)+P(B)-P(A \cup B) $
$ =\frac{3}{8}+\frac{5}{8}-\frac{3}{4}=\frac{2}{8}=1 / 4$
$P(A / B)=\frac{1 / 4}{5 / 8}=2 / 5$
$P\left(A^{'} / B\right)=\frac{P(A)-P(A \cap B)}{P(B)}=\frac{5 / 8-\frac{1}{4}}{5 / 8}=\frac{3}{5}$
$P(A / B) \cdot P\left(A^{1} / B\right)=\frac{2}{5} \cdot \frac{3}{5}=\frac{6}{25}$