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:
Answer & explanation
Correct answer: option 3
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,
$P(A) = \frac{2}{3} - \frac{2}{9} = \frac{4}{9}$