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)

From the definition of independence of events
$P(A|B)=\frac{P(A\cap B)}{P(B)}$
Then $P(B)\cdot P(A|B)=P(A\cap B)...(1)$
Interchanging the role of $A$ and $B$ in (1)
$P(A)P(B|A)=P(B\cap A)...(2)$
As $A\cap B=B\cap A$, we have from $(1)$ and $(2)$
$P(A)P(B|A)=P(B)P(A|B)$
$\Rightarrow \frac{1}{4}\cdot \frac{2}{3}=P(B)\cdot \frac{1}{2}$
$\Rightarrow P(B)=\frac{1}{4}\cdot \frac{2}{3}\cdot 2=\frac{1}{3}$