Let A and B be two events in a probability space with P(A) = 0.3, P(B) = 0.5, and P(A | B) = 0.1. Which of the following statements is/are TRUE? A. The two events A and B are independent and P(A | B) = 0.3 and P(B | A) = 0.5, respectively. B. P(A union B) = 0.75 C. P(A^c | B) = 0.9, where A^c is the complement of the event A D. P(A | B^c) = P(A | B), where A^c and B^c are the complements of the events A and B respectively

GATE 2024 · Engineering Mathematics · Conditional Probability and Independence · medium

Answer: Statements B and C are TRUE.

1. Find P(A and B): P(A and B) = 0.1 * 0.5 = 0.05.
2. Check Statement A: independence: P(A)*P(B) = 0.3*0.5 = 0.15 != 0.05 = P(A and B). NOT independent. Statement A is FALSE.
3. Check Statement B: P(A union B): P(A union B) = 0.3 + 0.5 - 0.05 = 0.75. Statement B is TRUE.
4. Check Statement C: P(A^c | B): P(A^c|B) = 1 - 0.1 = 0.9. Statement C is TRUE.
5. Check Statement D: P(A | B^c): P(A and B^c) = P(A) - P(A and B) = 0.3 - 0.05 = 0.25. P(B^c) = 0.5. P(A|B^c) = 0.25/0.5 = 0.5 != 0.1 = P(A|B). Statement D is FALSE.