Question

If the events $A$ and $B$ are independent and if $P\left(\overline{A}\right)=\frac{2}{3},P\left(\overline{B}\right)=\frac{2}{7}$, then $P(A\cap B)$ is equal to

• A. $\frac{4}{21}$
• B. $\frac{3}{21}$
• C. $\frac{5}{21}$
• D. $\frac{2}{21}$
• E. $\frac{1}{21}$

Answer 1

The correct option is C $\frac{5}{21}$

Given, $P\left(\overline{A}\right)=\frac{2}{3}$

Therefore, $P\left(A\right)=1-P\left(\overline{A}\right)=1-\frac{2}{3}=\frac{1}{3}$

and $P\left(\overline{B}\right)=\frac{2}{7}$

$\therefore P\left(B\right)=1-P\left(\overline{B}\right)=1-\frac{2}{7}=\frac{5}{7}$

Given, $A$ and $B$ are independent events.

Thus $P(A\cap B)=P\left(A\right)\cdot P\left(B\right)$

$=\frac{1}{3}\cdot \frac{5}{7}=\frac{5}{21}$