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Unformatted text preview: t A, P (A) ≤ 1. That is because if A is an event, then P (A) = 1 −
P (Ac ) ≤ 1 by Property p.4 and by the fact, from Axiom P.1, that P (Ac ) ≥ 0.
Property p.6 P (∅) = 0. That is because ∅ and Ω are complements of each other, so by Property
p.4 and Axiom P.3, P (∅) = 1 − P (Ω) = 0.
Property p.7 If A ⊂ B then P (A) ≤ P (B ). That is because B = A ∪ (Ac B ) and A and Ac B are
mutually exclusive, and P (Ac B ) ≥ 0, so P (A) ≤ P (A) + P (Ac B ) = P (A ∪ (Ac B )) = P (B ).
Property p.8 P (A ∪ B ) = P (A) + P (B ) − P (AB ). That is because, as illustrated in Figure 1.1,
A ∪ B can be written as the union of three mutually exclusive sets: A ∪ B = (AB c ) ∪ (Ac B ) ∪
(AB ). So
P (A ∪ B ) = P (AB c ) + P (Ac B ) + P (AB )
= (P (AB c ) + P (AB )) + (P (Ac B ) + P (AB )) − P (AB )
= P (A) + P (B ) − P (AB ).
Property p.9 P (A ∪ B ∪ C ) = P (A) + P (B ) + P (C ) − P (AB ) − P (AC ) − P (BC ) + P (ABC ).
This is a generalization of Property p....
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.
 Spring '08
 Zahrn
 The Land

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