Question

If$P(A)=P(B)=x$ and $P(A\cap B)=P(A^{\prime }\cap B^{\prime })=1/3$, then $x=?$

• A. $1/2$
• B. $1/3$
• C. $1/4$
• D. $1/6$

Answer 1

Answer: (A)

Since, $P\left(A^{\prime }\cap B^{\prime }\right)=\frac{1}{3}$
So, $P\left(\overline{A\cup B}\right)=\frac{1}{3}\Rightarrow 1-P\left(A\cup B\right)=\frac{1}{3}$
$\Rightarrow P\left(A\cup B\right)=\frac{2}{3}$
$\Rightarrow P\left(A\right)+P\left(B\right)=P\left(A\cap B\right)=\frac{2}{3}$
$\left(\because P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)\right)$
$\Rightarrow x+x-\frac{1}{3}=\frac{2}{3}\Rightarrow 2x=1\Rightarrow x=\frac{1}{2}$