Consider a random experiment where two fair coins are tossed. Let A be the event that denotes HEAD on both the throws, B be the event that denotes HEAD on the first throw, and C be the event that denotes HEAD on the second throw. Which of the following statements is/are TRUE? A. A and B are independent. B. A and C are independent. C. B and C are independent. D. Prob(B | C) = Prob(B)

GATE 2023 · Engineering Mathematics · Independent Events · medium

Answer: Options C and D are true: B and C are independent, and P(B|C) = P(B).

1. Compute individual and joint probabilities: P(A) = 1/4, P(B) = 2/4 = 1/2, P(C) = 2/4 = 1/2. P(A intersect B) = |{HH}|/4 = 1/4. P(A intersect C) = |{HH}|/4 = 1/4. P(B intersect C) = |{HH}|/4 = 1/4.
2. Test A and B for independence: P(A intersect B) = 1/4 != 1/8 = P(A)*P(B). So A and B are NOT independent. Option A is false.
3. Test A and C for independence (option B): P(A intersect C) = 1/4 != 1/8 = P(A)*P(C). So A and C are NOT independent. Option B is false.
4. Test B and C for independence (option C): P(B intersect C) = 1/4 = P(B)*P(C). So B and C ARE independent. Option C is true.
5. Check P(B | C) = P(B) (option D): P(B|C) = 1/2 = P(B). Option D is true.