[...]... chance of being the victor Consequently, each player 12 Introduction to Probability Theory has player probability 1/n of being the victor Now, suppose these n players are divided into r teams, with team i containing ni players, i = 1, , r That is, suppose players 1, , n1 constitute team 1, players n1 + 1, , n1 + n2 constitute team 2 and so on Then the probability that the victor is a member of... 688 691 696 697 703 706 707 710 715 717 722 723 723 725 726 734 Appendix: Solutions to Starred Exercises 735 Index 775 Preface This text is intended as an introduction to elementary probability theory and stochastic processes It is particularly well suited for those wanting to see how probability theory can be applied to the study of phenomena in fields such as engineering, computer science, management... Calculate the probability that the player wins 14 The probability of winning on a single toss of the dice is p A starts, and if he fails, he passes the dice to B, who then attempts to win on her toss They continue tossing the dice back and forth until one of them wins What are their respective probabilities of winning? 15 Argue that E = EF ∪ EF c , E ∪ F = E ∪ FEc 16 Use Exercise 15 to show that P(E... as the sample space of the experiment and is denoted by S Introduction to Probability Models, ISBN: 9780123756862 Copyright © 2010 by Elsevier, Inc All rights reserved 2 Introduction to Probability Theory Some examples are the following 1 If the experiment consists of the flipping of a coin, then S = {H, T} where H means that the outcome of the toss is a head and T that it is a tail 2 If the experiment... probability models Other possible courses would be a one-semester course in introductory probability theory (involving Chapters 1–3 and parts of others) or a course in elementary stochastic processes The textbook is designed to be flexible enough to be used in a variety of possible courses For example, I have used Chapters 5 and 8, with smatterings from Chapters 4 and 6, as the basis of an introductory... Henk Tijms, Vrije University Zhenyuan Wang, University of Binghamton Ward Whitt, Columbia University Bo Xhang, Georgia University of Technology Julie Zhou, University of Victoria xv This page intentionally left blank Introduction to Probability Theory 1.1 CHAPTER 1 Introduction Any realistic model of a real-world phenomenon must take into account the possibility of randomness That is, more often than... Suppose each of three persons tosses a coin If the outcome of one of the tosses differs from the other outcomes, then the game ends If not, then the persons start over and retoss their coins Assuming fair coins, what is the probability that the game will end with the first round of tosses? If all three coins are biased and have probability 1 of landing heads, what is the probability that the game will... exhibit an inherent variation that should be taken into account by the model This is usually accomplished by allowing the model to be probabilistic in nature Such a model is, naturally enough, referred to as a probability model The majority of the chapters of this book will be concerned with different probability models of natural phenomena Clearly, in order to master both the “model building” and the subsequent... research It is generally felt that there are two approaches to the study of probability theory One approach is heuristic and nonrigorous and attempts to develop in the student an intuitive feel for the subject that enables him or her to “think probabilistically.” The other approach attempts a rigorous development of probability by using the tools of measure theory It is the first approach that is employed... , denoted by ∞ En , is defined to be the event consisting of those outcomes that n=1 are in all of the events En , n = 1, 2, Finally, for any event E we define the new event Ec , referred to as the complement of E, to consist of all outcomes in the sample space S that are not in E That is, Ec will occur if and only if E does not occur In Example (4) 4 Introduction to Probability Theory if E = {(1, .