Axiom p3 p 1 2 in fact the choice of which properties

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Unformatted text preview: B2 ∪ · · · )c . Probability axioms The probability measure P is assumed to satisfy the following axioms: Axiom P.1 For any event A, P (A) ≥ 0. Axiom P.2 If A, B ∈ F and if A and B are mutually exclusive, then P (A ∪ B ) = P (A) + P (B ). More generally, if E1 , E2 , . . . is an infinite list (i.e. countably infinite collection) of mutually exclusive events, P (E1 ∪ E2 ∪ · · · ) = P (E1 ) + P (E2 ) + · · · . Axiom P.3 P (Ω) = 1. 2 In fact, the choice of which properties to make axioms is not unique. For example, we could have made Property e.4 an axiom instead of Axiom E.1. 1.2. AXIOMS OF PROBABILITY 9 If Axioms P.1-P.3 are satisfied (and Axioms E.1-E.3 are also satisfied) then the probability measure P has other intuitively reasonable properties. We list them here: Property p.4 For any event A, P (Ac ) = 1−P (A). That is because A and Ac are mutually exclusive events and Ω = A ∪ Ac . So Axioms P.2 and P.3 yield P (A) + P (Ac ) = P (A ∪ Ac ) = P (Ω) = 1. Property p.5 For any even...
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