Let A and B be two independent events such that $P(A) = \large\frac{1}{5} \: P( A \cup B ) = \large\frac{7}{10}$. Ten $P(\overline B)$ is equal to

$\begin {array} {1 1} (A)\;\large\frac{3}{8} & \quad (B)\;\large\frac{2}{7} \\ (C)\;\large\frac{7}{9} & \quad (D)\;None\: of \: these \end {array}$

Let $P( \overline B ) = x$ Then $P(B)=1-x$
$P(A \cap B)=P(A) . P(B)$
$= \large\frac{1}{5} (1-x)$
$P(A \cup B )=P(A)+P(B)-P(A \cap B )$
$\large\frac{7}{10} = \large\frac{1}{5} +(1-x) - \large\frac{1}{5} (1-x)$
$\large\frac{1}{2}=\large\frac{4}{5} (1-x)$
$1-x = \large\frac{5}{8}$
$x = \large\frac{3}{8}$
Ans : (A)