Question

If A and B are independent events of a random experiment such that $P(A\cap B)=\frac{1}{6}$ and $P(\overline{A}\cap \overline{B})=\frac{1}{3}$, then $P\left(A\right)$ is equal to

• A. $\frac{1}{3}$
• B. $\frac{2}{3}$
• C. $\frac{5}{7}$
• D. None of these

Answer 1

The correct option is A $\frac{1}{3}$

Given $P(A\cap B)=\frac{1}{6}$ and $P(\overline{A}\cap \overline{B})=\frac{1}{3}$

$P(\overline{A}\cap \overline{B})=P\left(\overline{A\cup B}\right)=1-P(A\cup B)$

$\Rightarrow P(A\cup B)=1-\frac{1}{3}=\frac{2}{3}$

We know that $P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$

$\Rightarrow P\left(A\right)+P\left(B\right)=\frac{2}{3}+\frac{1}{6}=\frac{5}{6}$

Given $A$ and $B$ are independent events , so we get $P\left(A\right)P\left(B\right)=P(A\cap B)=\frac{1}{6}$

By solving above equations , we get $P\left(A\right)=\frac{1}{3}$ and $P\left(B\right)=\frac{1}{2}$

Therefore option $A$ is correct.