Question

It is given that the events $A$ and $B$ are such that $P(A)=\frac{1}{4},P(A/B)=\frac{1}{2}$ and $P(B/A)=\frac{2}{3}$. Then, $P(B)$ is

• A. $\frac{1}{2}$
• B. $\frac{1}{6}$
• C. $\frac{1}{3}$
• D. $\frac{2}{3}$

Answer 1

Answer: (C)

Given that,
$P(A)=\frac{1}{4},P\left(\frac{A}{B}\right)=\frac{1}{2}\text{ and }P\left(\frac{B}{A}\right)=\frac{2}{3}$
We know that, $P\left(\frac{A}{B}\right)=\frac{P(A\cap B)}{P(B)}$...(i)
and $P\left(\frac{B}{A}\right)=\frac{P(B\cap A)}{P(A)}$...(ii)
$\therefore P(B)=\frac{P\left(\frac{B}{A}\right)\cdot P(A)}{P\left(\frac{A}{B}\right)}$
$=\frac{\left(\frac{2}{3}\right)\left(\frac{1}{4}\right)}{\left(\frac{1}{2}\right)}$
$=\frac{2}{3}\times \frac{1}{4}\times \frac{2}{1}=\frac{1}{3}$