Question

Three events A, B and C have probabilities $\frac{2}{5},\frac{1}{3}$ and $\frac{1}{2}$, respectively. If,

$P(A\cap C)=\frac{1}{5}andP(B\cap C)=\frac{1}{4}$, then find the values of $P\left(\frac{C}{B}\right)andP(A^{\prime }\cap C^{\prime }).$

Answer 1

Here, $P\left(A\right)=\frac{2}{5},P\left(B\right)=\frac{1}{3},P\left(C\right)=\frac{1}{2},P(A\cap C)=\frac{1}{5}andP(B\cap C)=\frac{1}{4}$
$\therefore P\left(\frac{C}{B}\right)=\frac{P(B\cap C)}{P\left(B\right)}=\frac{\frac{1}{4}}{\frac{1}{3}}=\frac{3}{4}andP(A^{\prime }\cap C^{\prime })=1-P(A\cup C)=1-\left[P\right(A)+P(C)-P(A\cap C\left)\right]=1-[\frac{2}{5}+\frac{1}{2}-\frac{1}{5}]=1-\left[\frac{4+5-2}{10}\right]=1-\frac{7}{10}=\frac{3}{10}$