Question

Let \( A \) and \( B \) be two events. If \( P(A) = 0.49 \), \( P(B) = 0.3 \) and \( P(A | B^c) = 0.4 \), then \( P(A | B) \) is equal to

• A. \( 0.45 \)
• B. \( 0.28 \)
• C. \( 0.4 \)
• D. \( 0.7 \)
• E. \( 0.3 \)

Hint:

Use the Law of Total Probability when dealing with conditional probabilities for complement events.

Answer 1

The Correct Option is D

Step 1: Law of Total Probability \[ P(A) = P(A | B) P(B) + P(A | B^c) P(B^c) \] 
where: 
- \( P(A) = 0.49 \) - \( P(A | B^c) = 0.4 \) 
- \( P(B) = 0.3 \), so \( P(B^c) = 1 - 0.3 = 0.7 \) 
Step 2: Solve for \( P(A | B) \) \[ 0.49 = P(A | B) (0.3) + (0.4)(0.7) \] \[ 0.49 = 0.3 P(A | B) + 0.28 \] \[ 0.3 P(A | B) = 0.49 - 0.28 = 0.21 \] \[ P(A | B) = \frac{0.21}{0.3} = 0.7 \] 
 Final Answer: \[ \boxed{0.7} \]