Question

Let $A$ and $B$ be events such that $P\left(A\right)-\frac{1}{6},P\left(B\right)=\frac{1}{4}$ and $P(A\cap B)=\frac{1}{12}$. Find $P\left(B/A\right).P\left(A/B\right)$.

Answer 1

We know that

$P\left(A/B\right)=\frac{P(A\cap B)}{P\left(B\right)}$ and $P\left(B/A\right)=\frac{P(A\cap B)}{P\left(A\right)}$

$\Rightarrow P\left(B/A\right).P\left(A/B\right)=\frac{P(A\cap B)}{P\left(B\right)}.\frac{P(A\cap B)}{P\left(A\right)}$

$\Rightarrow P\left(B/A\right).P\left(A/B\right)=\frac{\frac{1}{12}}{\frac{1}{4}}.\frac{\frac{1}{12}}{\frac{1}{6}}$

$\Rightarrow P\left(B/A\right).P\left(A/B\right)=\frac{1}{3}.\frac{1}{2}$

$\Rightarrow P\left(B/A\right).P\left(A/B\right)=\frac{1}{6}$