Question

If \( P(A | B) = P(A' | B) \), then which of the following statements is true?

• A. \( P(A) = P(A') \)
• B. \( P(A) = 2P(B) \)
• C. \( P(A \cap B) = \frac{1}{2}P(B) \)
• D. \( P(A \cap B) = 2P(B) \)

Hint:

Conditional probability uses the relationship \( P(A | B) = \frac{P(A \cap B)}{P(B)} \). Use complementarity to solve such problems.

Answer 1

The Correct Option is C

The condition \( P(A | B) = P(A' | B) \) implies: \[ \frac{P(A \cap B)}{P(B)} = \frac{P(A' \cap B)}{P(B)}. \] Simplify: \[ P(A \cap B) = P(A' \cap B). \] Since \( A \) and \( A' \) are complements: \[ P(A \cap B) + P(A' \cap B) = P(B). \] Substitute \( P(A \cap B) = P(A' \cap B) \): \[ P(A \cap B) + P(A \cap B) = P(B) \implies 2P(A \cap B) = P(B). \] Thus: \[ P(A \cap B) = \frac{1}{2}P(B). \] The correct answer is (C) \( P(A \cap B) = \frac{1}{2}P(B) \).