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Unformatted text preview: Chapter 4 Finite Probability Spaces In this chapter we briefly discuss some basic concepts from the theory of finite prob ability. The presentation is selfcontained, although it is assumed that readers have at least some basic idea of what is meant by a statement such as: “The probability of event A occurring is 1 3 ”. It is important to note that the presentation here is based on the assumption that the sample space is a finite set. Significant modifications are required if the sample space is an infinite set. 4.1 Sample Spaces, Events, and Probability Mea sures By a finite probability space we mean a pair (Ω , P ) where Ω is a nonempty finite set, called a sample space , and P is a nonnegative realvalued function on Ω, called a probability measure on Ω. The elements of Ω are called elementary events . For a given ω ∈ Ω, P ( ω ) is the probability that ω will occur. The probabilities of the elementary events must sum to 1, i.e. summationdisplay ω ∈ Ω P ( ω ) = 1 . (4.1) Although it will usually be the case that P ( ω ) > 0 for all ω ∈ Ω, we allow for the possibility that P ( ω ) = 0 for some elementary events ω ∈ Ω. Subsets of Ω are called events . If A ⊂ Ω, then the probability of A is defined to be P [ A ] = summationdisplay ω ∈ A P ( ω ) . (4.2) Here the notation “ A ⊂ B ” means that A is a subset of B and the possibility that A = B is allowed. We write “ A subsetnoteql B ” to indicate that A is a proper subset of B . Remark 4.1 . If A ⊂ Ω, we write P [ A ] to denote the probability of A , whereas if ω ∈ Ω we write P ( ω ) to denote the probability of ω . This distinction frequently is not made. However, certain formulas are clarified by having a way to know by inspection if the 115 inputs for P are elements of Ω or subsets of Ω. Notice that with this convention we have P ( ω ) = P [ { ω } ] for all ω ∈ Ω. We make the standard convention that P [ ∅ ] = 0, where ∅ is the empty set (or null set). Notice that P [Ω] = 1. If A ⊂ Ω and we denote A c = { ω ∈ Ω : ω / ∈ A } then P [ A c ] = 1 − P [ A ] because each ω ∈ Ω belongs either to A or A c , but not to both. More generally, if A,B ⊂ Ω, we have P [ A ∪ B ] = P [ A ] + P [ B ] − P [ A ∩ B ] . (4.3) (You are asked to verify (4.3) in Exercise 4.2(b) of this chapter.) Example 4.2. Consider a random experiment in which three coins are tossed. An elementary event can be described by a string of length three ( α,β,γ ), where α,β,γ ∈ { H,T } . Here “ H ” stands for heads and “ T ” stands for tails. The string ( α,β,γ ) represents the elementary event that the first coin shows α , the second coin shows β , and the third coin shows γ . There are 2 3 = 8 elementary events....
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 Spring '10
 SCHAFFER

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