Q. If $P(A \cap B) = \frac{1}{3},P(A \cup B) = \frac{5}{6}$ and $P(A) = \frac{1}{2}$, then which one of the following is correct?
A) $A$ and $B$ are independent events
B) $A$ and $B$ are mutually exclusive events
C) $P(A) = P(B)$
D) $P(A) < P(B)$
Explanation
(a)
$P(A \cup B) = P(A) + P(B) - P(A \cap B)$
$\Rightarrow \ \frac{5}{6} = \frac{1}{2} + P(B) - \frac{1}{3}$
$\Rightarrow \ P(B) = \frac{2}{3}$
$Also,\ P(A).P(B) = \frac{1}{2}.\frac{2}{3} = \frac{1}{3}$
$\therefore\ P(A \cap B) = P(A).P(B)$
Hence, $A$ and $B$ are independent events
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.