Question

If $P\left(A\right)=P\left(B\right)=x$ and $P(A\cap B)=P(A^{\prime }\cap B^{\prime })=\frac{1}{3}$, then $x$ is equal to

• A. $\frac{1}{2}$
• B. $\frac{1}{3}$
• C. $\frac{1}{4}$
• D. $\frac{1}{6}$

Answer 1

Answer: (A)

$\frac{1}{2}$

Given, $P\left(A\right)=P\left(B\right)=x$
and $P(A\cap B)=P(A^{\prime }\cap B^{\prime })=\frac{1}{3}$
$\therefore P(A^{\prime }\cap B^{\prime })=P{(A\cup B)}^{\prime }$
$\Rightarrow \frac{1}{3}=1-P(A\cup B)$
$\Rightarrow P(A\cup B)=1-\frac{1}{3}=\frac{2}{3}$
Now, $P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$
$\Rightarrow \frac{2}{3}=x+x-\frac{1}{3}$
$\Rightarrow \frac{2}{3}+\frac{1}{3}=2x$
$\Rightarrow x=\frac{1}{2}$