Question

If $A$ and $B$ are independent events such that $P\left(A\right)=\frac{1}{5}$, $P(A\cup B)=\frac{7}{10}$, then what is $P\left(\overline{B}\right)$ equal to?

• A. $\frac{2}{7}$
• B. $\frac{3}{7}$
• C. $\frac{3}{8}$
• D. $\frac{7}{9}$

Answer 1

Answer: (C)

$\frac{3}{8}$

Let $P\left(B\right)=x$

As $A$ and $B$ are independent events,

$P(A\cap B)=P\left(A\right)P\left(B\right)=\frac{x}{5}$

Now, $P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$

$\Rightarrow \frac{7}{10}=\frac{1}{5}+x-\frac{x}{5}$

$\Rightarrow \frac{1}{2}=\frac{4x}{5}$

$\Rightarrow x=\frac{5}{8}$

And, $P\left(\overline{B}\right)=1-x=1-\frac{5}{8}=\frac{3}{8}$

Hence, C is correct.