If A and B are independent events and $P(A) =\frac{1}{2}, P(B) =\frac{1}{3}$ then

Match List-I with List-II

Choose the correct answer from the options given below:

(A)-(III), (B)-(I), (C)-(IV), (D)-(II)

The correct answer is Option (1) → (A)-(III), (B)-(I), (C)-(IV), (D)-(II)

Given: $P(A)=\frac{1}{2},\ P(B)=\frac{1}{3}$ and A, B are independent.

Then $P(A\cap B)=P(A)P(B)=\frac{1}{6}$.

(A) $P(A\cap B)=\frac{1}{6}$ → (III)

(B) $P(\bar{A})P(B)+P(A)P(\bar{B}) = (1-\frac{1}{2})(\frac{1}{3}) + (\frac{1}{2})(1-\frac{1}{3}) = \frac{1}{6} + \frac{1}{3} = \frac{1}{2}$ → (I)

(C) $P(A|B)+P(B|A) = \frac{P(A\cap B)}{P(B)} + \frac{P(A\cap B)}{P(A)} = \frac{\frac{1}{6}}{\frac{1}{3}} + \frac{\frac{1}{6}}{\frac{1}{2}} = \frac{1}{2} + \frac{1}{3} = \frac{5}{6}$ → (IV)

(D) $P(A\cap\bar{B}) = P(A) - P(A\cap B) = \frac{1}{2} - \frac{1}{6} = \frac{1}{3}$ → (II)