Question

Three events $A$, B and $C$ have probabilities $\frac{2}{5},\frac{1}{3}$ and $\frac{1}{2}$ respectively. Given that $P(A\cap B)=\frac{1}{5}$ and $P(B\cap C)=\frac{1}{4},$ find the value of $P(C\mid B)$ and $P(A^{\prime }\cap C^{\prime })$

Answer 1

Here, $P\left(A\right)=\frac{2}{5},P\left(B\right)=\frac{1}{3},P\left(C\right)=\frac{1}{2}\cdot P(A\cap C)=\frac{1}{5}$ and $P(B\cap C)=\frac{1}{4}$

$\therefore P\left(C/B\right)=\frac{P(B\cap C)}{P\left(B\right)}=\frac{1/4}{1/3}=\frac{3}{4}$

And $P(A^{\prime }\cap C)=1-P(A\cup C)=1-\left[P\right(A)+P(C)-P(A\cap C\left)\right]$

$=1-[\frac{2}{5}+\frac{1}{2}-\frac{1}{5}]=1-\left[\frac{4+5-2}{10}\right]=1-\frac{7}{10}=\frac{3}{10}$