Question

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

• A. $\frac{3}{8}$
• B. $\frac{2}{7}$
• C. $\frac{7}{9}$
• D. None of these

Answer 1

The correct option is A $\frac{3}{8}$

Let $P\left(\overline{B}\right)=x$, then $P\left(B\right)=1-P\left(\overline{B}\right)=1-x$
$\therefore P(A\cap B)=P\left(A\right)\cdot P\left(B\right)=\frac{1}{5}(1-x)$
But $P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$
$\Rightarrow \frac{7}{10}=\frac{1}{5}+(1-x)-\frac{1}{5}(1-x)$
$\Rightarrow \frac{7}{10}-\frac{1}{5}=(1-\frac{1}{5})(1-x)$
$\Rightarrow \frac{5}{10}=\frac{4}{5}(1-x)$
$\Rightarrow (1-x)=\frac{5}{8}$
$\Rightarrow x=1-\frac{5}{8}$
$\therefore x=\frac{3}{8}$