Question

If $P\left(B\right)=\frac{3}{4},P(A\cap B\cap \overline{C})=\frac{1}{3}andP(\overline{A}\cap B\cap \overline{C})=\frac{1}{3}$, then $P(B\cap C)=\frac{\lambda }{\mu }$, where $\lambda$ and $\mu$ are coprime, then $\mu -5\lambda$ is

Answer 1

$P(B\cap \overline{C})=P[(A\cup \overline{A})\cap (B\cap \overline{C})]$
$P(A\cap B\cap \overline{C})+P(\overline{A}\cap B\cap \overline{C})$
$=\frac{1}{3}+\frac{1}{3}=\frac{2}{3}$
$P(B\cap C)=P\left(B\right)-P(B\cap \overline{C})=\frac{3}{4}-\frac{2}{3}=\frac{1}{12}$
$\Rightarrow \mu -5\lambda =7$