4 Algorithms: An algorithm is the abstract version of a computer program. In essence a computer program is the codification of an algorithm, and every algorithm can be coded as a computer program.

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29 Markov Chains: Markov chains are one of the most fun tools of probability; they give a lot of power for very little effort. We will restrict ourselves to finite Markov chains. This is not much of a restriction. We get much of the most interesting cases and avoid much of the theory.1 This chapter defines Markov chains and gives you one of the most basic tools for dealing with them: matrices.

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Introduction to Methods of Applied Mathematics: For the past few years I have been working on an open source textbook. It contains material on calculus, functions of a complex variable, ordinary differential equations, partial differential equations and the calculus of variations. The text is still under development, but I believe that the current version will be useful for students and instructors.

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37 A Little Game Theory: Game theory is one of the most interesting topics of discrete mathematics. The principal theorem of game theory is sublime and wonderful. We will merely assume this theorem and use it to achieve some of our early insights. To appreciate the theorem it is not necessary to know the proof. Do not let any math pedant tell you otherwise.1 By a game mathematics refers to a conflict between individuals (or entities) with conflicting goals.

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36 Poisson Processes: We are going to look at a random process that typifies what most of us think of as random. This process has the added virtues that it is easy to work with and it is used a great deal in mathematical modeling. In fact, it may be used more than any other process of its type. In particular what I am talking about is the Poisson1 process which is described by both the Poisson distribution and the exponential distribution.

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35 Mixed Chains: In this chapter we learn how to analyze Markov chains that consists of transient and absorbing states. Later we will see that this analysis extends easily to chains with (nonabsorbing) ergodic states.

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34 Ergodic Chains: Until this point in the book, results have been derived or proven, or at least the nature of the proof has been indicated. However, proofs of the techniques of Sections 34, 35 and 36 as well as the core concept of Chapter 37 are beyond this book and have been omitted. Do not let this deter you.

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33 The Gambler's Ruin: There are many variations of the gambler's ruin problem, but a principal one goes like this. We have i dollars out of a total of n. We flip a coin which has probability p of landing heads, and probability q = 1 ! p of landing tails. If the coin lands heads we gain another dollar, otherwise we lose a dollar.

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32 Averages: You should remember the arithmetic average. Given n data points, their arithmetic average is their sum divided by n. Now suppose that we have the average of n numbers, An. We are given a new data point x and we would like to compute the new average of all n + 1 numbers, An + 1. Many people simply add up all n + 1 numbers and then divide by n + 1. However,

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23 The Law of Multiplication: Probability trees are one of the most useful tools in probability and applied mathematics. We will use them throughout the rest of this text. The graph in Figure 1 is a probability tree.

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