Match List-I with List-II
If \( P(A) = \frac{3}{7},\; P(B) = \frac{4}{7} \) and \( P(A \cup B) = \frac{5}{7} \)
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List-I |
List-II |
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(A) \( P(A \cap B) \) |
(I) \( \frac{2}{3} \) |
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(B) \( P(A|B) \) |
(II) \( \frac{5}{7} \) |
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(C) \( P(B|A) \) |
(III) \( \frac{1}{2} \) |
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(D) \( P(A' \cup B') \) |
(IV) \( \frac{2}{7} \) |
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)
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List-I |
List-II |
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(A) \( P(A \cap B) \) |
(IV) \( \frac{2}{7} \) |
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(B) \( P(A|B) \) |
(III) \( \frac{1}{2} \) |
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(C) \( P(B|A) \) |
(I) \( \frac{2}{3} \) |
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(D) \( P(A' \cup B') \) |
(II) \( \frac{5}{7} \) |
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)