Question

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

• A. $5/12$
• B. $3/8$
• C. $5/8$
• D. $1/4$

Answer 1

Answer: (A)

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