# Introduction to Probability

Charles M. Grinstead, Swarthmore College

J. Laurie Snell, Dartmouth College

Copyright Year:

Publisher: American Mathematical Society

Language: English

## Formats Available

## Conditions of Use

Free Documentation License (GNU)

Free Documentation License (GNU)

## Reviews

This text provides very good coverage of the essential topics for an introductory probability course in addition to its coverage of topics that I’m sure are left out of some introductory courses such as Markov processes and generating functions. ... read more

This text provides very good coverage of the essential topics for an introductory probability course in addition to its coverage of topics that I’m sure are left out of some introductory courses such as Markov processes and generating functions. The strength of this book in my view (which is from an engineering perspective) is that it approaches topics in a very natural way, using practical examples, simple graphics, and discussion of computer simulation when introducing topics. It does not seem to burden the reader with statistical jargon or needlessly deep discussions of theory, but it does not give the impression that it is trying to avoid these things either. In my opinion, the book omits all the right things, including most of the tables found in introductory probability and statistics texts. The book is well-organized as any good textbook should be. The table of contents, index, and preface are all helpful.

While there are far too many examples and problems to check every one, I found no errors in the problems and examples I did work through. A closer review of what I consider to be essential content also revealed no errors. I would consider the content to be accurate.

The content is up-to-date and on-par with other books on the subject that are used in the engineering discipline. The addition of computer simulation examples does not detract from its relevance or longevity at all because it is approached in a very general manner, likely making transition between disciplines and over time easier. I use computer simulation when teaching probability, but would not use the programs used by the authors. This fact would not deter me from adopting the text.

The greatest strength of this book in my view is its clarity. The examples are presented in a logical way and the writing style is not a burden. It is also very concise, making it easy to digest the material. Formulae are presented simply (for an audience with the appropriate background) with clear explanations in the text.

The terminology, writing style, and logical development of concepts is consistent throughout. The coverage of topics also seems to be balanced, not favoring deeper treatment of only some topics. Other introductory texts I have experience with seem to trail-off when it comes to the more advanced topics.

The chapters seem to be compact and self-contained, making it possible to progress through the material in an order other than listed in the table of contents. The separation of chapter 10 (Generating Functions) and chapter 5 (Distribution and Densities) is a refreshing change from other textbooks I have used and helps with the modularity a great deal.

The flow of the book is very logical, but does present topics in a slightly different order than I would in a classroom. Fortunately, the modularity of the book is good, which gives instructors the freedom to choose the flow that works best for them.

I found no problems with the text, graphics, or other aspects. It just seems like a normal book. It does not have more advanced features that are common in “online” textbooks such as hyperlinks and imbedded content. I do not find this problematic, but some may.

I do not recall any grammatical errors in my reading nor did I make notes of any errors as I worked my way through the book.

The book is written in an accessible way. Its examples are relatable, but not trivial, and does not broach topics in a way that could be viewed as offensive. More importantly, understanding the examples does not seem like it would be predicated on having a deep understanding of a particular subject matter (e.g., engineering) or culture.

Of the probability and statistics books I have used, I consider this to be one of the better ones for explaining difficult probability concepts clearly. I prefer not to be constrained to a specific textbook (and therefore a specific style) when teaching a class. Happily, I feel this book would not be constraining at all and would support many teaching styles and instructional approaches to introductory probability. I especially like the numerous exercises. At a minimum, I intend to begin using this textbook as a reference in my course immediately, with the expectation of making it the primary textbook in the very near future.

The book covers all subjects that I need except the required materials on joint distributions. It would be great to have two more chapters to cover joint probability distributions for discrete and continuous random variables. Also I feel that the... read more

The book covers all subjects that I need except the required materials on joint distributions. It would be great to have two more chapters to cover joint probability distributions for discrete and continuous random variables. Also I feel that the last chapter on random walks is not necessary to be included.

It does seem to be free of errors

Yes, the content is up-to-date and the book with adding some materials on joint distributions is good to serve as an introduction of probability for undergraduates.

well written and with a lot of interesting examples and exercise problems.

yes, the terminology and framework in this book are consistent

I would like to move CH3 on combinatorics to the front before talking about any distribution in CH1 and CH2. Also CH5 on some important distributions could be split to CH1 and CH2.

The book is well organized but could be better with some changes, see my comments in Item 6 above

index terms on the back could be improved with some click-through function.

I do not see grammatical problem (I am not a native speaker)

no cultural issues found in the book

This is a good introduction book on probability, especially it is free to students. I hope that the authors could update the book soon with considering my suggestions

The book covers all areas in a typical introductory probability course. The course would be appropriate for seniors in mathematics or statistics or data science or computer science. It is also appropriate for first year graduate students in any... read more

The book covers all areas in a typical introductory probability course. The course would be appropriate for seniors in mathematics or statistics or data science or computer science. It is also appropriate for first year graduate students in any of these fields.

The book is very accurate.

Content is up-to-date. In fact, the way simulations are used to illustrate important concepts in probability and statistics

is now more relevant that ever ! the emerging focus on computing and computing-related areas like the field of Data Science and Data Analytics or Big Data makes this book and important textbook or resource. So, this is the right book or resource and No Need to Re-invent the Wheel!!!!

The book is very clear and smooth. Everything is classic or traditional except few places where I noticed a difference of what I am used to see: the authors used a unique notation, m(x), for the distribution function (cdf) in the discrete case compared to that for the continuous case. Also,I am not sure that the selected vector and complement notations are commonly used.

The book is consistent and the material flows nicely! the important concepts are introduced and revisited many times and sometimes different ways! I love the connection made with other areas! I love the use of Paradoxes.

Modularity is another major strength of the book! Although the material is nicely connected but but once can easily select to cover certain parts and skip others without creating gaps or difficulties in the students leering. The flow of the coverage and the nature of the probability area help in this matter.

You can easily treat or cover the discrete random variables separately and select the related material without any difficulties. You can do the same thing for the continuous case. You can leave some of the challenging examples that include some of the paradoxes that maybe challenging for students!

You can also easily and smoothly teach or assign the history and development of the selected topics as reading s without making it as a part of the graded course!

Overall, the material is presented in a smooth way! but that is not necessarily the order I would go with when i cover these topics.

Of course that is a matter of style, depending on the audience, I think it is easier to teach the material in Ch10 (moment generating functions), then may be add a section about movements. I would probably slightly modify it. I would point at few other things later!

The book is free of any interface issues.

No grammatical errors

The book is written with examples and problems that are very relevant to the culture we are in. Examples form the business world (examples include insurance coverage and insurance-related problems, gambling and lottery, sports, etc.)

Yes, I have specific comments that maybe useful to the authors:

First: Thank you:

Thank you for writing such a wonderful book. It is very clear, that the standards you held are really high and the timing of the book is unbelievable appropriate! with the new emerging statistical fields, this book should be used in the core courses!

Second: I have few specific suggestions/few typos that maybe useful. If you are interested, please let me know.

There is a table of contents that breaks up the chapters into subtopics, also. There is an index. Not much depth in some areas. There isn't much talked about with certain graphics, aka defining histograms and pie charts. Hypothesis testing is... read more

There is a table of contents that breaks up the chapters into subtopics, also. There is an index. Not much depth in some areas. There isn't much talked about with certain graphics, aka defining histograms and pie charts. Hypothesis testing is limited. There are no solutions in the back of the book to the chapter problems. Correlation? Probability is covered well. Statistics (aka Prob and Stats)?

The book does seem to be free of errors.

The book's relevance and longevity shouldn't be a problem. All information is relevant to the topic. I read the online version. So, one would think that any updates would be rather easily accomplished.

The book wasn't very clear for me. I noticed several times where, for example, individual cases were listed simply within the formatting of the paragraph, where these should probably be outlined, with bullet-points. Or, definitions are given within the framework of the paragraphs, where these should probably be separated from the paragraph, given their own spaces in the text. Even though important terms may be italicized, it can still be difficult to identify them within the readings. Some of the symbols, you have to take your time to make sure you understand just what it is describing.

The textbook does seem to be consistent in its use of terminology and framework. It does show consistent structure, rather than going "hodge podge" every once in a while.

Each chapter in the book does show suptopics on the table of contents. As for re-ordering the chapters, that may have something to do with how the individual instructor conducts the class. As in, if the instructor re-orders the material, they are probably going to have to provide some of their own introduction material for each chapter.

The flow tends to be a bit tedious at times. Some steps and/or terms are written in the format of the paragraphs, themselves, and not set apart from the rest of the writing.

The interface is decent. Some of the charts and tables are 2-4 pages off. But, the way the author did many of the graphics, he grouped many of the graphics together on certain pages. Computer programs are mentioned throughout the examples, but there are no computer codes or programs listed anywhere.

The book's grammar was fine. Very well written in this aspect.

There are seemingly no distinguishing cultural insensitivities.

I felt this textbook could do more. For the price, free, you can't beat it. However, considering as a student, to prepare me for future coursework and work on the job, I believe this textbook leaves much out. I remember taking a course with a book like this; I had to end up taking a separate Statistics class, also, because the course was certain statistics work. A lot of the symbolism comes up on you right away; you really have to take the time to understand the meaning of it. The missing information could be covered by a good instructor, but then there wouldn't be a need for a textbook in those parts. With as many times computer programs were referenced, it would have been nice to actually see the code for these programs at times, at least.

The book covers the fundamentals of probability theory with quite a few practical engineering applications, which seems appropriate for engineering students to connect the theory to the practice. Each chapter contains realistic examples that apply... read more

The book covers the fundamentals of probability theory with quite a few practical engineering applications, which seems appropriate for engineering students to connect the theory to the practice. Each chapter contains realistic examples that apply probability theory to basic statistical inference and naturally connect to the Monte Carlo simulations and graphical illustration of the probability distributions and probability density functions. Students with basic calculus and discrete math can easily follow the development of probabilistic modeling and important properties of the popularly used probability distributions. Only the ergodic Markov chain and random walk appear challenging for the undergraduate students to comprehend without the formal introduction of stochastic process.

I do no see any apparent error in the examples covered by this book.

The book can serve as an introduction of the probability theory to engineering students and it supplements the continuous and discrete signals and systems course to provide a practical perspective of signal and noise, which is important for upper level courses such as the classic control theory and communication system design. The material seems up-to-date and may be appealing to students with experience of Matlab to simulate various random events including the Markov chain and random walk covered in the later chapters.

The book is well written with many interesting exercise problems for students to enhance their understanding. It would be nice to provide a few solutions to the selected problems.

The terminology and framework used in the book are consistent.

The book can be divided and/or regrouped easily to fit the need for self-study. The discrete and continuous random variables and popularly used distributions can be summarized in separate tables (e.g., putting in the appendix).

The book is well organized with coherent logical development. It would be nice to add a brief introduction to continuous time and discrete time stochastic processes before introducing the Markov chain and random walk.

The book has many index terms but not available for click-through in the electronic format.

I do not see any grammatical problem.

I do not see any cultural issue in the examples used to demonstrate the probability theory.

This is a very nice introduction book to probability theory without using axiomatic and/or set theoretic coverage of the probability. It contains many interesting examples to demonstrate how to apply a probabilistic modeling or statistical procedure to study the real world phenomena. The integration with some software (such as Matlab) would provide better visualization of the random events, distribution and statistical properties of the random variable/process.

The book consists of 12 chapters, 3 appendices with tables and index. It is designed for an introductory probability course, for use in a standard one-term course, in which both discrete and continuous probability is covered. This book covers a... read more

The book consists of 12 chapters, 3 appendices with tables and index. It is designed for an introductory probability course, for use in a standard one-term course, in which both discrete and continuous probability is covered. This book covers a little bit more than I would normally cover in a probability class (Markov chains and random walks) and omits nothing that I would normally cover. All subject areas address in the Table of Contents are covered thoroughly.

The book is mathematically accurate as far as I can tell. Examples are worked out in full detail throughout the text. In the earlier version were some mistakes, but have been corrected (errata is available on the website). All of these errors have been corrected in the current web version.

The content is as up-to-date as any introductory probability textbook can reasonably be. In terms of longevity, the fact that the text of the book is stored in LaTeX ensures that the text will be useful for a long time to come. Updates will be straightforward to implement. There are over 600 exercises in the text. There are exercises to be done with and without the use of a computer and more theoretical exercises. A solution manual is available to instructors from website (odd-numbered exercices) or from the authors. In the text the computer is utilized in several ways: simulation, graphical illustration and to solve problems that do not lend to closed-form formulas. All programs used in the text have been written in TrueBASIC, Maple, and Mathematica.

I think that the text in this book is extremely clear, which is great for a first course in probability. It helps a large number of figures illustrating the discussed ideas. Authors have tried to present probability without too much formal mathematics but without sacrificing rigor. They have tried to develop the key ideas to provide a variety of interesting applications in normal live.

The text is consistent in its terminology, both internally and globally.

The text is divided into small subsections with separate exercices for students to read (there are easily be used as modules).

The organization is fine. The book presents all the topics in an appropriate sequence. I expect that instructors using this book would be using the material in the presented order (maybe Combinatorics first).

The interface is OK. I didn't experience any problems. The lack of color graphics even in digital version (few times authors use light blue color). I reviewed using the pdf version of the book. This does not have a linked table of contents, which would allow direct access to the sections. I wish the pdf file had this functionality. The lack of hyperlinks is somewhat troublesome.

I found no grammatical errors in this textbook (but English is not my native language). It is very well written.

No portion of this text appeared to me to be culturally insensitive or offensive in any way, shape, or form.

I think that this textbook provides a great introduction to probability! With such textbook available to students for free, I do not see any reasons to force my students to purchase a different textbook. My only complaint concerns the software. I would have preferred programs to be written in the language R.

There are numerous very interesting historical comments in the text.

## Table of Contents

- 1 Discrete Probability Distributions
- 2 Continuous Probability Densities
- 3 Combinatorics
- 4 Conditional Probability
- 5 Distributions and Densities
- 6 Expected Value and Variance
- 7 Sums of Random Variables
- 8 Law of Large Numbers
- 9 Central Limit Theorem
- 10 Generating Functions
- 11 Markov Chains
- 12 Random Walks

## Ancillary Material

## About the Book

Probability theory began in seventeenth century France when the two great French mathematicians, Blaise Pascal and Pierre de Fermat, corresponded over two problems from games of chance. Problems like those Pascal and Fermat solved continuedto influence such early researchers as Huygens, Bernoulli, and DeMoivre in establishing a mathematical theory of probability. Today, probability theory is a wellestablished branch of mathematics that finds applications in every area of scholarlyactivity from music to physics, and in daily experience from weather prediction topredicting the risks of new medical treatments.

This text is designed for an introductory probability course taken by sophomores,juniors, and seniors in mathematics, the physical and social sciences, engineering,and computer science. It presents a thorough treatment of probability ideas andtechniques necessary for a form understanding of the subject. The text can be usedin a variety of course lengths, levels, and areas of emphasis.

For use in a standard one-term course, in which both discrete and continuousprobability is covered, students should have taken as a prerequisite two terms ofcalculus, including an introduction to multiple integrals. In order to cover Chapter 11, which contains material on Markov chains, some knowledge of matrix theoryis necessary.

The text can also be used in a discrete probability course. The material has beenorganized in such a way that the discrete and continuous probability discussions arepresented in a separate, but parallel, manner. This organization dispels an overlyrigorous or formal view of probability and o?ers some strong pedagogical valuein that the discrete discussions can sometimes serve to motivate the more abstractcontinuous probability discussions. For use in a discrete probability course, studentsshould have taken one term of calculus as a prerequisite.

Very little computing background is assumed or necessary in order to obtain fullbenefits from the use of the computing material and examples in the text. All ofthe programs that are used in the text have been written in each of the languagesTrueBASIC, Maple, and Mathematica.

## About the Contributors

### Authors

**Charles M. Grinstead**, Professor, Department of Mathematics and Statistics, Swarthmore College.

**James Laurie Snell**, often cited as J. Laurie Snell, was an American mathematician. A graduate of the University of Illinois, he taught at Dartmouth College until retiring in 1995.