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## Über dieses Buch

DISCRETE PROBABILITY is a textbook, at a post-calculus level, for a first course in probability. Since continuous probability is not treated, discrete probability can be covered in greater depth. The result is a book of special interest to students majoring in computer science as well as those majoring in mathematics. Since calculus is used only occasionally, students who have forgotten calculus can nevertheless easily understand the book. The slow, gentle style and clear exposition will appeal to students. Basic concepts such as counting, independence, conditional probability, randon variables, approximation of probabilities, generating functions, random walks and Markov chains are presented with good explanation and many worked exercises. An important feature of the book is the abundance of problems, which students may use to master the material. The 1,196 numerical answers to the 405 exercises, many with multiple parts, are included at the end of the book. Throughout the book, various comments on the history of the study of probability are inserted. Biographical information about some of the famous contributors to probability such as Fermat, Pascal, the Bernoullis, DeMoivre, Bayes, Laplace, Poisson, Markov, and many others, is presented. This volume will appeal to a wide range of readers and should be useful in the undergraduate programs at many colleges and universities.

## Inhaltsverzeichnis

### Chapter 1. Introduction

Abstract
Everyone knows that, when a coin is tossed, the probability of heads is one-half. The hard question is, “What does that statement mean?” A good part of the discussion of that point would fall under the heading of philosophy and would be out of place here. To cover the strictly mathematical aspects of the answer, we need merely state our axioms and then draw conclusions from them. We want to know, however, why we are using the word “probability” in our abstract reasoning. And, more important, we should know why it is reasonable to expect our conclusions to have applications in the real world. Thus we do not want to just do mathematics. When a choice is necessary, we shall prefer concrete examples over mathematical formality. Before we get to mathematics, we present some background discussion of the question we raised a moment ago.
Hugh Gordon

### Chapter 2. Counting

Abstract
Recall that we saw in the last chapter that determining probabilities often comes down to counting how many this-or-thats there are. In this chapter we gain practice in doing that. While occasionally we shall phrase problems in terms of probability, the solutions will always be obtained by counting, until, in later chapters, we develop more probabilistic methods.
Hugh Gordon

### Chapter 3. Independence and Conditional Probability

Abstract
In this chapter we study how the knowledge that one event definitely occurs influences our judgement as to the chances for some other event to occur. The case we consider first is that in which there is no influence.
Hugh Gordon

### Chapter 4. Random Variables

Abstract
Often the results of an experiment are quantitative, rather than qualitative; in other words, the outcome of the experiment is a number or numbers. Assume for the moment that the result of the experiment is a single number, say the amount of money we shall be paid as the result of a bet. Then we may want to take into account that, if we receive $50, that is exactly half of$100. For some purposes, we regard a probability of one-fifth of getting $100 as equivalent to a probability of two-fifths of getting$50. We next introduce random variables into our study of probability to make it possible to take a quantitative point of view along those lines.
Hugh Gordon

### Chapter 5. More About Random Variables

Abstract
The last chapter covered the basic facts about random variables. In this chapter we discuss a number of different topics related only in that all involve random variables. The first section covers some very important theoretical matters that are necessary for understanding the basic ideas of probability theory. The second section discusses the computation of expected values in certain circumstances; this material is used extensively in Chapters Eight and Nine. The optional third section is concerned with finding variances. The three sections may be read in any order.
Hugh Gordon

### 6 Chapter. Approximating Probabilities

Abstract
Sometimes an approximation is more valuable than an exact answer. For example, consider the following question: A certain association has 4000 members; 2000 of them are men and 2000 are women. Exactly half the members will be randomly chosen to each receive one ticket to a special event; all members have the same chance of getting a ticket. What is the probability that exactly 1000 men and exactly 1000 women get tickets? The answer to the question is quite obviously
$$\frac{{{{\left( {\begin{array}{*{20}{c}} {2000} \\ {1000} \end{array}} \right)}^2}}}{{\left( {\begin{array}{*{20}{c}} {4000} \\ {2000} \end{array}} \right)}}$$
Hugh Gordon

### 7 Chapter. Generating Functions

Abstract
In this short chapter, we introduce a certain method of treating those random variables that, like most of those we study, take only non-negative integers as values. This method is useful in more general circumstances, and it will be easier to understand if we first present it in its natural setting. What we want to introduce can properly be called a tool, a trick, a toy, or a technique. A more helpful word is code. By describing a sequence of numbers in what appears to be a roundabout way, we often make it easier to work with the sequence.
Hugh Gordon

### 8 Chapter. Random Walks

Abstract
While this chapter is entitled “Random Walks,” random walks will be mentioned by name only in a few paragraphs. Superficially, the chapter seems to be about the story of two characters named Peter and Paul. Actually, what is involved is an abstract idea that may be made concrete in two different ways. The idea is most effectively put into words by discussing Peter and Paul. For this reason, we discuss Peter and Paul. In thinking about the problems we face, a picture in which a “particle” is actually “walking” about is often helpful. We shall explain how to imagine such a picture. A visualization of the moving particle is helpful; a verbal description of it is not. In any case, the Peter-and-Paul approach has historical interest.
Hugh Gordon

### 9 Chapter. Markov Chains

Abstract
In this chapter we generalize the situation considered in the last chapter. Recall that in the last chapter Peter and Paul were gambling. At any time during their play, their finances were in a certain condition or state. The word state is the one usually used here. Thus we had a “system” that could be in any one of a number of states. In fact, it moved from state to state as time went on. The motion was in discrete steps; each step corresponded to one game. We shall retain those concepts from the last chapter. We assumed that the amount bet was fixed at one dollar. Thus at each step Peter either gained a dollar or lost a dollar. We now want to generalize and allow the possibility of change in a single step from any state to any state.
Hugh Gordon

### Backmatter

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