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  4. A and B are events such that P(A∪ B)=3/4, P(A∩ B)=1/4, P( overlineA)=2/3 then P( overlineA∩ B) is

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Q. A and B are events such that $P\left(A\cup B\right)=3/4, P\left(A\cap B\right)=1/4, P\left(\overline{A}\right)=2/3$ then $P\left(\overline{A}\cap B\right)$ is

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Solution:

$P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right);$
$\Rightarrow \frac{3}{4}=1-P\left(\overline{A}\right)+P\left(B\right)-\frac{1}{4}$
$\Rightarrow 1=1-\frac{2}{3}+P\left(B\right) \Rightarrow P\left(B\right)=\frac{2}{3};$
Now, $P\left(\overline{A}\cap B\right)=P\left(B\right)-P\left(A\cap B\right)=\frac{2}{3}-\frac{1}{4}=\frac{5}{12}$.