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If A, B, C are three events then show that P (A ∪ B ∪ C) = P(A) + P(B) + P(C) – P (A ∩ B) – P(B ∩ C) – P(C ∩ A) + P(A ∩ B ∩ C) |
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Answer» Write B ∪ C = D then P(A ∪ B ∪ C) = P(A ∪ D) ∴ P(A ∪ D) = P(A) + P(D) – P(A ∩ D) = [P(A) + P(B ∪ C) – P(A ∩ (B ∪ C)] = P(A) + P(B) + P(C) – P(B ∩ C) – [P(A ∩ B) ∪ (A ∩ C)] = P(A) + P(B) + P(C) – P (B ∩ C) – [P(A ∩ B) + P(A ∩ C) – P(A ∩ B ∩ D ∩ C] P(A ∪ B ∪ C) = P(A) + P(B) + P(C) – P(A ∩ B) – P(B ∩ C) – P (C ∩ A) + P(A ∩ B ∩ C). |
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