**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)

$\large\frac{P}{3}=p+\frac{1}{3}-\frac{5}{9}$

$p-\large\frac{P}{3}=\frac{5}{9}-\frac{1}{3}$

$\large\frac{2p}{3}=\frac{2}{9}$

$p=\large\frac{1}{3}$