If A and B are two events such that $P(A/B) = p$, $P(A) = p$, $P(B) = \frac{1}{3}$ and $P(A \cup B) = \frac{5}{9}$, then $p$ is equal to…………..

Here, $P(A) = p,P(B) = \frac{1}{3}$

and $P(A \cup B) = \frac{5}{9}$

$P(A/B) = \frac{{P(A \cap B)}}{{P(B)}} = p \Rightarrow P(A \cap B) = \frac{p}{3}$

and $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

$\Rightarrow$ $\frac{5}{9} = p + \frac{1}{3} - \frac{p}{3} \Rightarrow \frac{5}{9} - \frac{1}{3} = \frac{{2p}}{3}$

$\Rightarrow$ $\frac{{5 - 3}}{9} = \frac{{2p}}{3} \Rightarrow p = \frac{2}{9} \times \frac{3}{2} = \frac{1}{3}$