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  4. If P (A) = P (B) = x and P(A∩ B)=P(A ' ∩ B')=1/3, then x=?

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Q. If$ P (A) = P (B) = x$ and $P(A\cap B)=P(A ' \cap B')=1/3$, then $x=?$

Probability

Solution:

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