For two events A and B, P(A)=P(A|B)=0.25 and P(B|A)=0.5. Which of the following are correct? I. A and B are independent. II. P(A^{c}\cup B^{c})=0.875 III. P(A^{c}\cap B^{c})=0.375 Select the answer using the code given below.

  1. A. I and II only
  2. B. II and III only
  3. C. I and III only
  4. D. I, II and III ✓

Correct Answer: D. I, II and III

Explanation

Since P(A|B) = P(A), events A and B are independent, making statement I true. Because they are independent, P(B|A) = P(B) = 0.5. Then P(A \cap B) = P(A)P(B) = (0.25)(0.5) = 0.125. By De Morgan's Law, P(A^c \cup B^c) = 1 - P(A \cap B) = 1 - 0.125 = 0.875 (Statement II true). Also, P(A^c \cap B^c) = P((A \cup B)^c) = 1 - (0.25 + 0.5 - 0.125) = 1 - 0.625 = 0.375 (Statement III true).

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