Match List-I with List-II

If \( P(A) = \frac{3}{7},\; P(B) = \frac{4}{7} \) and \( P(A \cup B) = \frac{5}{7} \) 

Choose the correct answer from the options given below:

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

The correct answer is Option (3) → (A) - (IV), (B) - (III), (C) - (I), (D) - (II)

Given:

$P(A) = \frac{3}{7}, \quad P(B) = \frac{4}{7}, \quad P(A \cup B) = \frac{5}{7}$

Using the identity:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

$\Rightarrow \frac{5}{7} = \frac{3}{7} + \frac{4}{7} - P(A \cap B)$

$\Rightarrow P(A \cap B) = \frac{3 + 4 - 5}{7} = \frac{2}{7}$

(A) $P(A \cap B) = \frac{2}{7}$ → (IV)

(B) $P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{2/7}{4/7} = \frac{1}{2}$ → (III)

(C) $P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{2/7}{3/7} = \frac{2}{3}$ → (I)

(D) $P(A' \cup B') = 1 - P(A \cap B) = 1 - \frac{2}{7} = \frac{5}{7}$ → (II)