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Unformatted text preview: ae Review of Probability Here we will review some of the basic facts usually taught in a first course in probability, concentrating on the ones that are used more than once in one of the six other chapters. This chapter may be read for review or skipped and referred to later if the need arises. 1. Probabilities, Independence We begin with a vague but useful definition. (Here and in what follows, bold face indicates a word or phrase that is being defined or explained.) The term experiment is used to refer to any process whose outcome is not known in advance. Two simple experiments are flip a coin, and roll a die. The sample space associated with an experiment is the set of all possible outcomes. The sample space is usually denoted by Ω, the capital Greek letter Omega. Example 1.1. Flip three coins. The flip of one coin has two possible outcomes, called “Heads” and “Tails,” and denoted by H and T . Flipping three coins leads to 2 3 = 8 outcomes: HHT HTT HHH HTH THT TTT THH TTH Example 1.2. Roll two dice. The roll of one die has six possible outcomes: 1, 2, 3, 4, 5, and 6. Rolling two dice leads to 6 2 = 36 outcomes { ( m,n ) : 1 ≤ m,n ≤ 6 } . The goal of probability theory is to compute the probability of various events of interest. Intuitively, an event is a statement about the outcome of an experiment. Formally, an event is a subset of the sample space. An example for flipping three coins is “two coins show Heads,” or A = { HHT,HTH,THH } 2 Review of Probability An example for rolling two dice is “the sum is 9,” or B = { (6 , 3) , (5 , 4) , (4 , 5) , (3 , 6) } Events are just sets, so we can perform the usual operations of set theory on them. For example, if Ω = { 1 , 2 , 3 , 4 , 5 , 6 } , A = { 1 , 2 , 3 } , and B = { 2 , 3 , 4 , 5 } , then the union A ∪ B = { 1 , 2 , 3 , 4 , 5 } , the intersection A ∩ B = { 2 , 3 } , and the complement of A , A c = { 4 , 5 , 6 } . To introduce our next defintion, we need one more notion: two events are disjoint if their intersection is the empty set, ∅ . A and B are not disjoint, but if C = { 5 , 6 } , then A and C are disjoint. A probability is a way of assigning numbers to events that satisfies: (i) For any event A , 0 ≤ P ( A ) ≤ 1. (ii) If Ω is the sample space, then P (Ω) = 1. (iii) For a finite or infinite sequence of disjoint events P ( ∪ i A i ) = ∑ i P ( A i ). In words, the probability of a union of disjoint events is the sum of the proba bilities of the sets. We leave the index set unspecified since it might be finite, P ( ∪ k i =1 A i ) = k X i =1 P ( A i ) or it might be infinite, P ( ∪ ∞ i =1 A i ) = ∑ ∞ i =1 P ( A i ). In Examples 1.1 and 1.2, all outcomes have the same probability, so P ( A ) =  A  /  Ω  where  B  is short for the number of points in B . For a very general example, let Ω = { 1 , 2 ,...,n } ; let p i ≥ 0 with ∑ i p i = 1; and define P ( A ) = ∑ i ∈ A p i . Two basic properties that follow immediately from the definition of a probability are...
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 Spring '09
 Lippitt
 Probability, Probability theory, probability density function

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