Probability

An Introduction

Samuel Goldberg

• Publisher: Dover Publications
• Year: 2013
• Print ISBN: 9780486652528
• eBook ISBN: 9780486318547
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

Excellent basic text covers set theory, probability theory for finite sample spaces, binomial theorem, probability distributions, means, standard deviations, probability function of binomial distribution, and other key concepts and methods essential to a thorough understanding of probability. Designed for use by math or statistics departments offering a first course in probability. 360 illustrative problems with answers for half. Only high school algebra needed. Chapter bibliographies.

Chapter 1

SETS

1. Examples of sets; basic notation

The concept of a set, whose fundamental role in mathematics was first pointed out in the work of the mathematician Georg Cantor (1845–1918), has significantly affected the structure and language of modern mathematics. In particular, the mathematical theory of probability is now most effectively formulated by using the terminology and notation of sets. For this reason, we devote Chapter 1 to the elementary mathematics of sets. Additional topics in set theory are included throughout the text, as the need for this material becomes apparent.

The notion of a set is sufficiently deep in the foundation of mathematics to defy being defined (at the level of this book) in terms of still more basic concepts. Hence, we can only aim here, by taking advantage of the reader’s knowledge of the English language and his experience with the real and conceptual world, to make clear the denotation of the word “set.”

A set is merely an aggregate or collection of objects of any sort: people, numbers, books, outcomes of experiments, geometrical figures, etc. Thus, we can speak of the set of all integers, or the set of all oceans, or the set of all possible sums when two dice are rolled and the number of dots on the uppermost faces are added, or the set consisting of the cities of Cambridge and Oberlin and all their residents, or the set of all straight lines (in a given plane) which pass through a given point.

The collection of objects must be well-defined, by which we mean that, for any object whatsoever, the question “Does this object belong to the collection?” has an unequivocal “yes” or “no” answer. It is not necessary that we personally have the knowledge required to decide which answer is correct. We must know only that, of the answers “yes” and “no,” exactly one is correct.

Let us also agree that no object in a set is counted twice; i.e., the objects are distinct. It follows that, when listing the objects in a set, we do not repeat an object after it is once recorded. For example, according to this convention, the set of letters in the word “banana” is a set containing not six letters, but rather the three distinct letters b, a and n.

The following definition summarizes our discussion to this point and introduces some additional terminology and notation.

Definition 1.1. A set is a well-defined collection of distinct objects. The individual objects that collectively make up a given set are called its elements, and each element belongs to or is a member of or is contained in the set. If a is an object and A a set, then we write a ∈ A as an abbreviation for “a is an element of A.” and a ∉ Λ for “a is not an element of A.” If a set has a finite number of elements, then it is called...

Table of contents

• Cover
• Title Page
• Copyright Page
• Preface
• Acknowledgments
• Contents
• Chapter 1. SETS
• Chapter 2. Probability in Finite Sample Spaces
• Chapter 3. Sophisticated Counting
• Chapter 4. Random Variables
• Cahpter 5. Binomial Distribution and some Applications
• Answers to Odd–Numbered Problems
• Index