Let A and B be any two arbitrary events, then which one of the following is TRUE? A. P(A ∩ B) = P(A) * P(B) B. P(A ∪ B) = P(A) + P(B) C. P(A|B) = P(A ∩ B) / P(A) D. P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

GATE 1994 · Engineering Mathematics · Conditional Probability · medium

Answer: D is always true: P(A ∪ B) = P(A) + P(B) - P(A ∩ B). Answer: D.

1. Eliminate incorrect options: A: P(A ∩ B) = P(A)*P(B) only if independent. Counterexample: A = B = any event with P(A) in (0,1): P(A ∩ A) = P(A) != P(A)^2. Wrong. B: P(A ∪ B) = P(A)+P(B) only if disjoint. Counterexample: take A=B: P(A ∪ A)=P(A) != 2P(A). Wrong. C: P(A|B) = P(A ∩ B)/P(A) is incorrect; definition requires denominator P(B). Wrong.
2. Verify option D: inclusion-exclusion: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) holds always by inclusion-exclusion. When counting outcomes in A ∪ B, outcomes in A ∩ B are counted twice in P(A)+P(B) and must be subtracted once.