Q. If $A$ and $B$ are two events, such that $P(A \cup B) = \frac{3}{4},\ P(A \cap B) = \frac{1}{4},P\left( \overline{A} \right) = \frac{2}{3}$, then $P(\overline{A} \cap B)$ is equal to
A) $\frac{5}{12}$
B) $\frac{3}{8}$
C) $\frac{5}{8}$
D) $\frac{1}{2}$
Explanation
(a)
$P(A) = 1 - P\left( \overline{A} \right) = 1 - \frac{2}{3} = \frac{1}{3}$
Using, $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
$\Rightarrow \frac{3}{4} = \frac{1}{3} + P(B) - \frac{1}{4} \Rightarrow P(B) = \frac{2}{3}$
$Now,\ P\left( \overline{A} \cap B \right) = (B) - P(A \cap B) = \frac{2}{3} - \frac{1}{4} = \frac{5}{12}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.