Question

If \( P(A \mid B) = P(A' \mid B) \), 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. \( P(A \cap B) = 2P(B) \)
• E.

Hint:

When dealing with conditional probabilities, use the definitions \( P(A \mid B) = \frac{P(A \cap B)}{P(B)} \) and \( P(A') = 1 - P(A) \) to simplify and solve equations.

Answer 1

The Correct Option is C

The conditional probability \( P(A \mid B) \) is defined as: \[ P(A \mid B) = \frac{P(A \cap B)}{P(B)}. \] Similarly: \[ P(A' \mid B) = \frac{P(A' \cap B)}{P(B)}. \] We are given that: \[ P(A \mid B) = P(A' \mid B). \] Substitute the definitions of conditional probabilities: \[ \frac{P(A \cap B)}{P(B)} = \frac{P(A' \cap B)}{P(B)}. \] Since \( P(B) > 0 \), cancel \( P(B) \) from both sides: \[ P(A \cap B) = P(A' \cap B). \] Using the property of probabilities: \[ P(A \cap B) + P(A' \cap B) = P(B). \] Substitute \( P(A \cap B) = P(A' \cap B) \) into the equation: \[ P(A \cap B) + P(A \cap B) = P(B). \] \[ 2P(A \cap B) = P(B). \] Solve for \( P(A \cap B) \): \[ P(A \cap B) = \frac{1}{2}P(B). \] Hence, the correct answer is (C) \( P(A \cap B) = \frac{1}{2}P(B) \).