Question

A, B are the events in a random experiment. If \( P(A)=\frac{1}{2}, P(B)=\frac{1}{3}, P(A \cap B)=\frac{1}{4} \), then \( P(A^c | B^c) + P(A | B) = \)

• A. 1
• B. 4/5
• C. 11/8
• D. 7/3

Hint:

Remember \( P(A^c \cap B^c) = 1 - P(A \cup B) \).

Answer 1

The Correct Option is C

Step 1: Calculate Probabilities:

\( P(A|B) = \frac{1/4}{1/3} = 3/4 \). \( P(A \cup B) = 1/2 + 1/3 - 1/4 = 7/12 \). \( P(A^c \cap B^c) = 1 - 7/12 = 5/12 \). \( P(A^c | B^c) = \frac{5/12}{1 - 1/3} = \frac{5/12}{2/3} = 5/8 \). Sum: \( 3/4 + 5/8 = 6/8 + 5/8 = 11/8 \).
Step 2: Final Answer:

11/8.