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Example 14 - Probability of atleast one of A, B is 1 - P(A') P(B')

Example 14 - Chapter 13 Class 12 Probability - Part 2

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Example 14 If A and B are two independent events, then the probability of occurrence of at least one of A and B is given by 1– P(A′) P(B′) Two events A and B are independent if P(A ∩ B) = P(A) . P(B) Probability of occurrence of at least one of A and B = Probability of occurrence of only A + Probability of occurrence of only B + Probability of occurrence of A and B = Probability of occurrence of A or B So, P(at least one of A & B) = P(A ∪ B) = P(A) + P(B) – P(A ∩ B) = P(A) + P(B) – P(A ∩ B) = P(A) + P(B) – P(A) . P(B) = P(A) + P(B) [ 1 – P(A) ] = P(A) + P(B) . P(A’) = 1 – P(A’) + P(B) . P(A’) = 1 – P(A’) [ 1 – P(B) ] = 1 – P(A’) . P(B’) ∴ P(at least one of A & B) = 1 – P(A’) P(B’) Hence Proved Given, A & B are independent So, P(A ∩ B) = P(A) . P(B)

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.