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 \( P(A \mid B) = P(A' \mid B) \). Substituting the definitions, we get \( \frac{P(A \cap B)}{P(B)} = \frac{P(A' \cap B)}{P(B)} \). Since \( P(B)>0 \), we can cancel \( P(B) \), yielding \( P(A \cap B) = P(A' \cap B) \). Using the probability property \( P(A \cap B) + P(A' \cap B) = P(B) \), and substituting \( P(A \cap B) = P(A' \cap B) \), we have \( P(A \cap B) + P(A \cap B) = P(B) \), which simplifies to \( 2P(A \cap B) = P(B) \). Solving for \( P(A \cap B) \) gives \( P(A \cap B) = \frac{1}{2}P(B) \). Thus, the correct answer is (C) \( P(A \cap B) = \frac{1}{2}P(B) \).