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:

When \( P(A|B) = P(A'|B) \), use the complement rule and the fact that probabilities sum to 1 to simplify.

Answer 1

The Correct Option is C

Given \( P(A|B) = P(A'|B) \), we use the definitions of conditional probability: \[ P(A|B) = \frac{P(A \cap B)}{P(B)}, \quad P(A'|B) = \frac{P(A' \cap B)}{P(B)}. \] Since \( A \) and \( A' \) are complementary events: \[ P(A \cap B) + P(A' \cap B) = P(B). \] From \( P(A|B) = P(A'|B) \), we have: \[ \frac{P(A \cap B)}{P(B)} = \frac{P(A' \cap B)}{P(B)}. \] This implies: \[ P(A \cap B) = P(A' \cap B). \] Using \( P(A \cap B) + P(A' \cap B) = P(B) \): \[ P(A \cap B) + P(A \cap B) = P(B). \] \[ 2P(A \cap B) = P(B). \] \[ P(A \cap B) = \frac{1}{2}P(B). \] Final Answer: \[ \boxed{P(A \cap B) = \frac{1}{2}P(B)} \]