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State and prove addition theorem of probability for two nonmutually exclusive events.

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Statement: Let A and B be any two events (Subsets of sample spaces) with respective probability P(A) and P(B). Then, the probability of occurence of atleast one of these events is: P(A ∪ B) = P(A) + P(B) – P(A ∩ B) where P(A ∩ B) is the simultaneous occurence of A and B.

Proof: A random experiment results in ‘n’ exhaustive outcomes out of which movt comes are favourable to event ‘A’, m2 outcomes are favourable to event ‘B’ and ‘p’ outcomes are common to both A and B. Consider the venn diagram:

The event of occurrence of at least one out of A and B is (A ∪ B) has (m1 + m2 – p) favourable outcomes.

Substituting the values from (1) P(A ∪ B) = P(A) + P(B) – P(A ∩B) Q.E.D.

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