If $A$ and $B$ are two events such that $P\left(\frac{A}{B}\right) = 2 \times P\left(\frac{B}{A}\right)$ and $P(A) + P(B) = \frac{2}{3}$, then $P(A)$ is equal to:

$\frac{4}{9}$

The correct answer is Option (3) → $\frac{4}{9}$ ##

$P\left(\frac{A}{B}\right) = 2 \times P\left(\frac{B}{A}\right)$

$\frac{P(A \cap B)}{P(B)} = \frac{2 \times P(A \cap B)}{P(A)}$

$P(A) = 2P(B)$

Given:

$P(A) + P(B) = \frac{2}{3}$

Substituting $P(A) = 2P(B)$:

$2P(B) + P(B)= \frac{2}{3}$

$3P(B) = \frac{2}{3}$

$P(B) = \frac{2}{9}$

Then,

$P(A) = \frac{2}{3} - \frac{2}{9} = \frac{4}{9}$