# If $P(A)=P(B)=x\: and \: P(A \cap B ) = P(A' \cap B')=\large\frac{1}{3}$ then x is equal to

$\begin {array} {1 1} (A)\;\large\frac{1}{2} & \quad (B)\;\large\frac{1}{3} \\ (C)\;\large\frac{1}{4} & \quad (D)\;\large\frac{1}{6} \end {array}$

Given that $P(A)=P(B)=x$
and $P(A \cap B )=P(A' \cap B')$
$= \large\frac{1}{3}$
$\therefore P(A'\cap B')=1- P(A \cup B )$
$= \large\frac{1}{3}$
$\Rightarrow P(A \cup B ) = 1- \large\frac{1}{3}$
$= \large\frac{2}{3}$
Now, $P(A \cup B ) = P(A)+P(B)-P(A \cap B )$
$\large\frac{2}{3}=x+x- \large\frac{1}{3}$
$2x= \large\frac{2}{3}+ \large\frac{1}{3}$
2x=1
$x= \large\frac{1}{2}$
Ans : (A)

answered Jan 16, 2014