Match List-I with List-II
If \(\mathrm{P(A)} = \displaystyle\frac{3}{7}, \mathrm{P(B)} = \frac{4}{7}\) and \(\mathrm{P(A \cup B)} = \displaystyle\frac{5}{7}\)
| List-I | List-II | ||
|---|---|---|---|
| (A) | \(\mathrm{P(A \cap B)}\) | (I) | \(\displaystyle\frac{2}{3}\) |
| (B) | \(\mathrm{P(A|B)}\) | (II) | \(\displaystyle\frac{5}{7}\) |
| (C) | \(\mathrm{P(B|A)}\) | (III) | \(\displaystyle\frac{1}{2}\) |
| (D) | \(\mathrm{P(A' \cup B')}\) | (IV) | \(\displaystyle\frac{2}{7}\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) - (IV), (B) - (III), (C) - (I), (D) - (II)
(A) \(P(A \cap B)\)
This matches (IV) \( \frac{2}{7} \). By the addition rule, \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\):
\( \displaystyle P(A \cap B) = \frac{3}{7} + \frac{4}{7} - \frac{5}{7} = \frac{2}{7} \)
(B) \(P(A|B)\)
This matches (III) \( \frac{1}{2} \). By the definition of conditional probability:
\( \displaystyle P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{2/7}{4/7} = \frac{1}{2} \)
(C) \(P(B|A)\)
This matches (I) \( \frac{2}{3} \):
\( \displaystyle P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{2/7}{3/7} = \frac{2}{3} \)
(D) \(P(A' \cup B')\)
This matches (II) \( \frac{5}{7} \). By De Morgan’s law, \(A' \cup B' = (A \cap B)'\):
\( \displaystyle P(A' \cup B') = 1 - P(A \cap B) = 1 - \frac{2}{7} = \frac{5}{7} \)
So the matching is (A) - (IV), (B) - (III), (C) - (I), (D) - (II).