# If A and B are two events such that $P(A|B)=p,P(A)=p,P(B)=\frac{1}{3} \quad$ and $\quad P(A\cup B)= \frac{5}{9},$ then $\; p=\text{_____________}.$?

Toolbox:
• If A and B are two events associated with an experiment.
• $P(A\cup B)=P(A)+P(B)-P(A\cap B)$
• $P(A/B)=\Large\frac{P(A\cap B)}{P(B)}$
Given
$P(A/B)=p$
$P(A)=p$
$P(B)=\large\frac{1}{3}$
$P(A\cap B)=\large \frac{5}{9}$
$P(A\cap B)=P(A)+P(B)-P(A\cup B)$ (1)
$P(A/B)=\Large\frac{p(A\cap B)}{P(B)}$
$P(A\cap B)=P(A/B) P(B)$
$P(A\cap B)=p\times\large\frac{1}{3}=\frac{P}{3}$ (2)
substituing values in (1)
$\large\frac{P}{3}=p+\frac{1}{3}-\frac{5}{9}$
$p-\large\frac{P}{3}=\frac{5}{9}-\frac{1}{3}$
$\large\frac{2p}{3}=\frac{2}{9}$
$p=\large\frac{1}{3}$