Isye 2027

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online