Question

If \( A \) and \( B \) are any two events of a random experiment and \( P(B) \neq 1 \), then: \[ P(A|B^C) = ? \]

• A. \( \frac{P(A) + P(A \cap B)}{1 - P(B)} \)
• B. \( \frac{P(A) - P(A \cap B)}{1 - P(B)} \)
• C. \( \frac{P(A) + P(A \cap B)}{1 + P(B)} \)
• D. \( \frac{P(A)}{1 + P(B)} \)

Hint:

When dealing with conditional probability and complements, use the identity \( P(A \cap B^C) = P(A) - P(A \cap B) \).

Answer 1

The Correct Option is B

Step 1: Use the definition of conditional probability. The conditional probability \( P(A|B^C) \) is given by: \[ P(A|B^C) = \frac{P(A \cap B^C)}{P(B^C)} \]
Step 2: Use the fact that \( P(A \cap B^C) = P(A) - P(A \cap B) \) and \( P(B^C) = 1 - P(B) \). Thus: \[ P(A|B^C) = \frac{P(A) - P(A \cap B)}{1 - P(B)} \]