Question

State the following statement is true or false

If $A$ and $B$ are independent events of a random experiment such that $P(\overline{A}\cap \overline{B})=\frac{1}{3}$ and $P(A\cap B)=\frac{1}{6}$, then $P\left(A\right)$ is equal to $\frac{1}{2}$ (Here, $E$ is the complement of the event $E$)

• A. True
• B. False

Answer 1

The correct option is A True

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

Since $A$ and $B$ are independent.

Therefore, $P\left(A\right)P\left(B\right)=\frac{1}{6}$ and $P\left(\overline{A}\right)P\left(\overline{B}\right)=\frac{1}{3}$

$\Rightarrow [1-P\left(A\right)][1-P\left(B\right)]=\frac{1}{3}$

$\Rightarrow 1-[P\left(A\right)+P\left(B\right)]+P\left(A\right)P\left(B\right)=\frac{1}{3}$

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

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

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