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

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Q. 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

Probability - Part 2

Solution:

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)
$\therefore P(B)=\frac{P\left(\frac{B}{A}\right) \cdot P(A)}{P\left(\frac{A}{B}\right)}$
$=\frac{\left(\frac{2}{3}\right)\left(\frac{1}{4}\right)}{\left(\frac{1}{2}\right)}$
$=\frac{2}{3} \times \frac{1}{4} \times \frac{2}{1}=\frac{1}{3}$