IMFC4 - Chapter 4 Finite Probability Spaces In this chapter...

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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 self-contained, 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 non-negative real-valued 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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IMFC4 - Chapter 4 Finite Probability Spaces In this chapter...

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