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 $\frac{3}{8}$

$\because P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)=P\left(A\right)+P\left(B\right)-P\left(A\right)P\left(B\right)$ since A and B are independent
putting the values
$\Rightarrow \frac{7}{10}=\frac{1}{5}+P\left(B\right)-\frac{P\left(B\right)}{5}\Rightarrow \frac{4P\left(B\right)}{5}=\frac{7}{10}-\frac{1}{5}=\frac{1}{2}\Rightarrow P\left(B\right)=\frac{5}{8}\Rightarrow P\left(B^c\right)=1-P\left(B\right)=1-\frac{5}{8}=\frac{3}{8}$