If A and B are independent events of a random experiment such that $P(A ∩ B) =\frac{1}{6}$ and $P(\overline{A} ∩ \overline{B}) =\frac{1}{3}$, then P(A) is equal to

$\frac{1}{3}$

We have,

$⇒P(A ∩ B) =\frac{1}{6}$ and $P(\overline{A} ∩ \overline{B}) =\frac{1}{3}$

$⇒P(A)P(B)=\frac{1}{6}\, and \, P(\overline{A}) P(\overline{B}) =\frac{1}{3}$

$⇒ xy =\frac{1}{6}\, and \, (1-x)(1-y)=\frac{1}{3},$ where $ P(A)= x, P(B) = y$

$⇒xy =\frac{1}{6} $ and $ 1-x -y +\frac{1}{6}=\frac{1}{3}$

$⇒xy =\frac{1}{6}$ and $ x + y= \frac{5}{6}$

$⇒xy =\frac{1}{6} $ and $y =\frac{1}{3}$ or $ x =\frac{1}{3}$ and $ y = \frac{1}{2}$