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A First Course in Probability
This introduction presents the mathematical theory of probability for readers in the fields of engineering and the sciences who possess knowledge of elementary calculus. Presents new examples and exercises throughout. Offers a new section that presents an elegant way of computing the moments of random variables defined as the number of events that occur. Gives applications to binomial, hypergeometric, and negative hypergeometric random variables, as well as random variables resulting from coupon collecting and match models. Provides additional results on inclusion-exclusion identity, Poisson paradigm, multinomial distribution, and bivariate normal distribution A useful reference for engineering and science professionals.
565 pages, Hardcover
First published January 1, 1976
112 people are currently reading
1376 people want to read
About the author
Sheldon M. Ross
77 books32 followersSheldon M. Ross is the Epstein Chair Professor at the Department of Industrial and Systems Engineering, University of Southern California. He received his Ph.D. in statistics at Stanford University in 1968 and was formerly a Professor at the University of California, Berkeley, from 1976 until 2004. He has published more than 100 articles and a variety of textbooks in the areas of statistics and applied probability, including Topics in Finite and Discrete Mathematics (2000), Introduction to Probability and Statistics for Engineers and Scientists, 4th edition (2009), A First Course in Probability, 8th edition (2009), and Introduction to Probability Models, 10th edition (2009), among others. Dr Ross serves as the editor for Probability in the Engineering and Informational Sciences.
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Displaying 1 - 30 of 38 reviews
October 15, 2020
Honestly, I hated this book with a passion, primarily due to the author's convoluted delivery style.
To enjoy this book you either need to
(i) have to a high pain tolerance
(ii) have to be sufficiently experienced with relevant mathematical topics to the extent that you can leverage that experience to QUICKLY fill the myriad massive gaps Sheldon leaves in his "explanations"
(iii) have a lot of time on your hands
If one randomly picks any 10 explanations or examples from the book, with a high probability, the expected value of the author's delivery is suboptimal. In many examples, the author solves problems in a manner that could only be done by someone who already knows the answer or has developed experience with the newly introduced concept (contradictory)
Here's something you'll find in the book:
let's say Sheldon wants to find A
he'll first obtain an expression/value for B, C, D & E
and then tell you that
A = B.C + D/E
as opposed to doing something like this:
A = B.C + D/E (because of so and so axiom or principle or proof)
B is ... (because of so and so axiom or principle or proof)
C is .... (because of so and so axiom or principle or proof)
D is ... (because of so and so axiom or principle or proof)
E ... (because of so and so axiom or principle or proof)
so plugging A,B,C & D them into the equation A = B.C + D/E
gives A = ...
I recommend, that Sheldon considers rewriting most of the examples in the book by starting from fundamental truths or axioms that are known, and following a coherent path to the answer. Perhaps a video version of the book would be better.
There are positives, in that it comprehensive (with respect to topics), and there are many practice problems
To enjoy this book you either need to
(i) have to a high pain tolerance
(ii) have to be sufficiently experienced with relevant mathematical topics to the extent that you can leverage that experience to QUICKLY fill the myriad massive gaps Sheldon leaves in his "explanations"
(iii) have a lot of time on your hands
If one randomly picks any 10 explanations or examples from the book, with a high probability, the expected value of the author's delivery is suboptimal. In many examples, the author solves problems in a manner that could only be done by someone who already knows the answer or has developed experience with the newly introduced concept (contradictory)
Here's something you'll find in the book:
let's say Sheldon wants to find A
he'll first obtain an expression/value for B, C, D & E
and then tell you that
A = B.C + D/E
as opposed to doing something like this:
A = B.C + D/E (because of so and so axiom or principle or proof)
B is ... (because of so and so axiom or principle or proof)
C is .... (because of so and so axiom or principle or proof)
D is ... (because of so and so axiom or principle or proof)
E ... (because of so and so axiom or principle or proof)
so plugging A,B,C & D them into the equation A = B.C + D/E
gives A = ...
I recommend, that Sheldon considers rewriting most of the examples in the book by starting from fundamental truths or axioms that are known, and following a coherent path to the answer. Perhaps a video version of the book would be better.
There are positives, in that it comprehensive (with respect to topics), and there are many practice problems
March 20, 2019
For a 'first course in probability', it's quite heavy and complete. Be ready for some mild headaches on the way since the book is exercise-centric. But it's worth it! If you want to learn virtually everything about combinations and random variables, this book is great. And I'm so glad to have learned about the concept of surprise and how, from a simple set of axioms, one can retrieve the entropy formula!! (check section 9.3). Another plus, despite the tremendous amount of exercises, all solutions are also provided (in like 80 pp. or so).
February 6, 2022
3.5
کتابیه که بحث های احتمال دانشگاهی رو پوشش می ده. لحن راحت و شیرینی داره و هر کس می تونه به تنهایی بخونتش و بفهمتش. برای کسایی که علاقمند به هوش مصنوعی هستند کتاب مفیدیه چون پیش نیاز هوش مصنوعیه. همین طور به افراد خارج از این حوزه، میتونه دید جدیدی بده و دریچه جدیدی رو باز کنه. مباحث احتمال به طور کلی در زندگی بسیار کاربرد دارند.
تنها ایرادی که میتونم بگیرم اینه که خیلی اوقات مسایل رو مفهومی بیان نمی کنه و صرفا از طریق ریاضی اون ها رو اثبات می کنه. در حالی که بیان شهودی فهم رو برای خواننده بهتر می کنه.
اگر خواستید به سراغش برید توصیه میکنم سراغ ترجمهاش نرید که زبانش سخته و حذفیاتش زیاده. انگلیسیش روان تره.
کتابیه که بحث های احتمال دانشگاهی رو پوشش می ده. لحن راحت و شیرینی داره و هر کس می تونه به تنهایی بخونتش و بفهمتش. برای کسایی که علاقمند به هوش مصنوعی هستند کتاب مفیدیه چون پیش نیاز هوش مصنوعیه. همین طور به افراد خارج از این حوزه، میتونه دید جدیدی بده و دریچه جدیدی رو باز کنه. مباحث احتمال به طور کلی در زندگی بسیار کاربرد دارند.
تنها ایرادی که میتونم بگیرم اینه که خیلی اوقات مسایل رو مفهومی بیان نمی کنه و صرفا از طریق ریاضی اون ها رو اثبات می کنه. در حالی که بیان شهودی فهم رو برای خواننده بهتر می کنه.
اگر خواستید به سراغش برید توصیه میکنم سراغ ترجمهاش نرید که زبانش سخته و حذفیاتش زیاده. انگلیسیش روان تره.
May 11, 2013
As the saying goes, probability is inherently challenging, not trivial like calculus. If your feelings about calculus are a bit different, you might find this book not so much a "first course" as a confusing deathmarch.
This ISBN is the 2nd Edition.
This ISBN is the 2nd Edition.
February 26, 2021
Eu tenho pouca disciplina para estudar um tema técnico sozinho. Tinha tendo estudar esse livro por conta uma vez, mas parei rápido. Dessa vez usei o livro para acompanhar um curso de probabilidade que fiz no Verão do IME e fluiu muito bem. Carrot and stick.
O livro é muito bem escrito e, ao meu ver, tem a forma de apresentação adequada para o tópico. Ou seja, focado na resolução de problemas. Não porque você precisa ser treinado para um teste, mas porque é impossível discutir probabilidade, em nível introdutório, sem exemplos concretos.
Lembro que quando tentei ler pela primeira vez, eu estranhei a dificudade que tinha para resolver problemas que me pareciam simples. Mas fazendo o curso de verão, na companhia de pessoas que se formaram em matématica e estatística, notei que minha dificuldade era comum. Na verdade, em matemática é comum um problema ter uma descrição simples e uma solução complexa. Isso acontece com muita frequência em probabilidade. Alguns problemas que o Ross apresenta no decorrer do livro são, na verdade, problemas que demoraram décadas ou até séculos para serem resolvidos. Então se na sua primeira leitura de um capítulo você não conseguir resolver muitos problemas, não se preocupe. Probabilidade é um tema difícil mesmo.
Recomendo.
O livro é muito bem escrito e, ao meu ver, tem a forma de apresentação adequada para o tópico. Ou seja, focado na resolução de problemas. Não porque você precisa ser treinado para um teste, mas porque é impossível discutir probabilidade, em nível introdutório, sem exemplos concretos.
Lembro que quando tentei ler pela primeira vez, eu estranhei a dificudade que tinha para resolver problemas que me pareciam simples. Mas fazendo o curso de verão, na companhia de pessoas que se formaram em matématica e estatística, notei que minha dificuldade era comum. Na verdade, em matemática é comum um problema ter uma descrição simples e uma solução complexa. Isso acontece com muita frequência em probabilidade. Alguns problemas que o Ross apresenta no decorrer do livro são, na verdade, problemas que demoraram décadas ou até séculos para serem resolvidos. Então se na sua primeira leitura de um capítulo você não conseguir resolver muitos problemas, não se preocupe. Probabilidade é um tema difícil mesmo.
Recomendo.
October 25, 2022
Good examples, good problems. Sometimes the explanations didn't help as much as I'd like them to, and sometimes the problems were overly-repetitive. Still, good.
May 10, 2024
The top rated review gives a positive view of the book and points out that it is 'quite heavy and complete.' A lot many users seem to have a negative experience with the book but still mention it being 'maths heavy.' I think the book falls in an area where it is too non rigorous for any student of maths (A purely combinatorial approach is taken in the exercises and Kolmogorov's axioms are forgotten as soon as they are written for the most part) but perhaps the details were too less for non maths folks.
Readers are advised to look for another book more suitable to their own mathematical maturity rather than this one.
Readers are advised to look for another book more suitable to their own mathematical maturity rather than this one.
June 20, 2021
3.0 out of 5 stars
Probably a Good Book for the Brilliant
July 31, 2005
While in college almost 25 years ago, I vowed never again to read a book that talks about pulling balls from urns: i.e., no more probability books. But, since this book is used in a required course for a degree program I was reading through (Florida State University's (FSU's) STA 4442: "Introductory Probability I" course, required for their Computer Science degree), I decided to try the subject again. I should have listened to myself and stayed away. I managed to get through the first four chapters before having to put the book away. There are just too many assumptions of knowledge and leaps of logic over vast quantities of missing steps for me to work my way through the book on my own. This is especially bad, since the Preface states:
"This book is intended as an elementary introduction to the theory of probability for students in mathematics, statistics, engineering, and the sciences (including computer science, the social sciences and management science) who posses the prerequisite knowledge of elementary calculus."
The first use of that word, "elementary," above, definitely doesn't belong there. Also, I shudder to think of the reaction of people in the social and management sciences who end up in a course using this book. On a more constructive note, I'd add to the prerequisites a *heavy* grounding in discrete math.
To add a bit more information to the raw data of these reviews, I've mapped the universe of all possible readers of this book onto a set of x-y axes. Let the x-axis run from "non-Math-types" up through "Math-types." Let they y-axis go from "non-geniuses" up through "geniuses:"
- Quadrant I: genius Math-types. This group, along with the Quadrant IIs, has a chance at getting something useful out of the book. It's fairly evenly divided between theory and application, but I'd say the emphasis appears to lean slightly towards using probability theory over proving it (only slightly, mind you). Everything, however, starts out with the theory and with proofs. The problem is that the proofs are missing the majority of their steps and require quite a bit of mathematical knowledge and bashing one's head onto the table to get through. Quadrant Is will probably have the least trouble with this.
- Quadrant II: genius non-Math-types. Like the Quadrant Is, above, these people have a chance at getting something useful out of the book. Though there is quite a bit of theory present, it's also coupled with quite a lot of applications via the examples (several for each theoretical blurb). Since there are so many steps missing from both the proofs and the examples, those on the "genius" side of the coordinate system are the ones most likely to have a chance at forcing themselves through the material to actual understanding.
- Quadrant III: non-genius non-Math-types (i.e., "normal" people). This group is going to have a lot of trouble with this book. With a good professor or knowledgeable person nearby, they might be able to work through the assumptions and gaps in the material (with a lot of effort). But, in general, this book is really not for them. The large number of examples will help, but won't be sufficient.
- Quadrant IV: non-genius Math-types. This group falls into the same boat as the Quadrant IIIs: the book's not for them. Though their math predilection might give them a moderate advantage in working out what the author is trying to do, they're still going to have trouble working through the proofs with all their missing steps.
This book is really only suitable for either genius Math-, or genius non-Math-types. I'd recommend that anyone who doesn't read through mathematics like fish swim through the sea avoid it. With a professor handy to fill in the gaps the author leaves, the book might be acceptable to us mortals. But, for those on their own, there's no way they'll get through. Because of the mismatch between the audience the Preface lays out and the way the text is written, I rate this book at 3 stars out of 5.
Probably a Good Book for the Brilliant
July 31, 2005
While in college almost 25 years ago, I vowed never again to read a book that talks about pulling balls from urns: i.e., no more probability books. But, since this book is used in a required course for a degree program I was reading through (Florida State University's (FSU's) STA 4442: "Introductory Probability I" course, required for their Computer Science degree), I decided to try the subject again. I should have listened to myself and stayed away. I managed to get through the first four chapters before having to put the book away. There are just too many assumptions of knowledge and leaps of logic over vast quantities of missing steps for me to work my way through the book on my own. This is especially bad, since the Preface states:
"This book is intended as an elementary introduction to the theory of probability for students in mathematics, statistics, engineering, and the sciences (including computer science, the social sciences and management science) who posses the prerequisite knowledge of elementary calculus."
The first use of that word, "elementary," above, definitely doesn't belong there. Also, I shudder to think of the reaction of people in the social and management sciences who end up in a course using this book. On a more constructive note, I'd add to the prerequisites a *heavy* grounding in discrete math.
To add a bit more information to the raw data of these reviews, I've mapped the universe of all possible readers of this book onto a set of x-y axes. Let the x-axis run from "non-Math-types" up through "Math-types." Let they y-axis go from "non-geniuses" up through "geniuses:"
- Quadrant I: genius Math-types. This group, along with the Quadrant IIs, has a chance at getting something useful out of the book. It's fairly evenly divided between theory and application, but I'd say the emphasis appears to lean slightly towards using probability theory over proving it (only slightly, mind you). Everything, however, starts out with the theory and with proofs. The problem is that the proofs are missing the majority of their steps and require quite a bit of mathematical knowledge and bashing one's head onto the table to get through. Quadrant Is will probably have the least trouble with this.
- Quadrant II: genius non-Math-types. Like the Quadrant Is, above, these people have a chance at getting something useful out of the book. Though there is quite a bit of theory present, it's also coupled with quite a lot of applications via the examples (several for each theoretical blurb). Since there are so many steps missing from both the proofs and the examples, those on the "genius" side of the coordinate system are the ones most likely to have a chance at forcing themselves through the material to actual understanding.
- Quadrant III: non-genius non-Math-types (i.e., "normal" people). This group is going to have a lot of trouble with this book. With a good professor or knowledgeable person nearby, they might be able to work through the assumptions and gaps in the material (with a lot of effort). But, in general, this book is really not for them. The large number of examples will help, but won't be sufficient.
- Quadrant IV: non-genius Math-types. This group falls into the same boat as the Quadrant IIIs: the book's not for them. Though their math predilection might give them a moderate advantage in working out what the author is trying to do, they're still going to have trouble working through the proofs with all their missing steps.
This book is really only suitable for either genius Math-, or genius non-Math-types. I'd recommend that anyone who doesn't read through mathematics like fish swim through the sea avoid it. With a professor handy to fill in the gaps the author leaves, the book might be acceptable to us mortals. But, for those on their own, there's no way they'll get through. Because of the mismatch between the audience the Preface lays out and the way the text is written, I rate this book at 3 stars out of 5.
January 30, 2025
I'm a Math PhD. I've read hundreds of math books and textbooks, and this is definitely in the top-tier. It teaches by means of examples and has close to a thousand exercises in total. Thus, you will learn by doing and building intuition. The theory is clearly and succintly presented and the range of topics is broad. It is similar in style to Strang's Introduction to Linear Algebra.
Ocasionally the author omits a proof or technicality, but this is rare and done with excellent judgement. The only criticism I have is that a few examples are difficult or too long, but overall this book is a pedagogical success. As a bonus, it contains a very concise introduction to (Shannon's) information theory.
I felt compelled to leave a review because a rating < 4 stars for this book is just obscene. For readers interested in probability I would also recommend Feller's book, who was a giant in this area. However, the latter is an old book and is definitely not a suitable as Ross' book as an introduction.
Ocasionally the author omits a proof or technicality, but this is rare and done with excellent judgement. The only criticism I have is that a few examples are difficult or too long, but overall this book is a pedagogical success. As a bonus, it contains a very concise introduction to (Shannon's) information theory.
I felt compelled to leave a review because a rating < 4 stars for this book is just obscene. For readers interested in probability I would also recommend Feller's book, who was a giant in this area. However, the latter is an old book and is definitely not a suitable as Ross' book as an introduction.
May 4, 2025
Look, it's a classic. Others have written about how Ross's explanations are fairly convoluted. which is true. I was surprised in the manner he chose to present definitions, which were often difficult to read with little consistency. I was surprised for the level of mathematical sophistication required for the book, which I had no difficulty with as a former graduate student in mathematics and current graduate student in artificial intelligence. However, I don't think I would have done well to read this book earlier in my education. And certainly, I don't think it does well as "a first course in probability" as the name states.
Otherwise, I read it to freshen up for my knowledge because I'm taking the Society of Actuaries Probability examination in a couple months. For that purpose it did fine, not the best book I've read in the field but not the worst either. More like "a good refresher on probability" to be quite honest, than anything of an introduction.
Otherwise, I read it to freshen up for my knowledge because I'm taking the Society of Actuaries Probability examination in a couple months. For that purpose it did fine, not the best book I've read in the field but not the worst either. More like "a good refresher on probability" to be quite honest, than anything of an introduction.
June 1, 2021
The name "a first course in probability" is a bit misleading. It might actually be accurate for math majors, but not for the general audience. The book presumes some knowledge in single and multivariate calculus, so be aware of that.
The book is amazing and if you have some math background you should be fine, altough you might have some difficulties in the proof sections. Take a look on mathematical induction and you'll be fine.
The best part of the book is that it's packed with examples. You have soooooo much applications on basically every topic in probability, and also full solutions for many exercises in each chapter. This is a great addition for self-learners and those who are reviewing the material by themselves.
Overall, you can't be wrong with that book, altough maybe it's name should be "a second course in probability"
The book is amazing and if you have some math background you should be fine, altough you might have some difficulties in the proof sections. Take a look on mathematical induction and you'll be fine.
The best part of the book is that it's packed with examples. You have soooooo much applications on basically every topic in probability, and also full solutions for many exercises in each chapter. This is a great addition for self-learners and those who are reviewing the material by themselves.
Overall, you can't be wrong with that book, altough maybe it's name should be "a second course in probability"
April 29, 2025
I hate this book with every neuron in my brain and every ounce of academic stamina I once had.
Sheldon Ross somehow pulled off the dark magic of turning one of the most fascinating topics in all of mathematics—Probability—into a soul-sucking vortex of despair. His writing is as dry as unbuttered toast and twice as hard to digest. There's no didactic clarity, no engaging tone, no spark of curiosity—just a slow, grinding descent into a statistical purgatory. And yet, for reasons I can only attribute to some long-standing academic curse, this book remains a staple in undergraduate Statistics courses. It's not a textbook. It's an endurance test.
Sheldon Ross somehow pulled off the dark magic of turning one of the most fascinating topics in all of mathematics—Probability—into a soul-sucking vortex of despair. His writing is as dry as unbuttered toast and twice as hard to digest. There's no didactic clarity, no engaging tone, no spark of curiosity—just a slow, grinding descent into a statistical purgatory. And yet, for reasons I can only attribute to some long-standing academic curse, this book remains a staple in undergraduate Statistics courses. It's not a textbook. It's an endurance test.
March 29, 2019
Mathematicians might like it; I don’t. The explanations are clumsy and not elucidating. No matter how smart the speaker might think oneself, the proof lies in the ability to convey information to a diverse audience with various backgrounds in a clear and concise way. Effectively.
Or, put another way: Explain it to me like I know nothing.
For a ‘first course’, this book does the job poorly.
My kid’s high school text does a better job with every single one of these concepts better.
Or, put another way: Explain it to me like I know nothing.
For a ‘first course’, this book does the job poorly.
My kid’s high school text does a better job with every single one of these concepts better.
April 17, 2023
A sufficient book to pair with a university course. I found it too often fell back on "present a fact/theorem/&c. and prove it" without explanation, justification or illumination. Maybe too focused on being rigorous and correct for the math major, and just hoping the science major will absorb the information with hundreds of different dry examples.
January 5, 2018
There are many good, challenging problems in this book (such as ___, ___, ___ and ____ (TO UPDATE LATER)).
I'm taking off one star though, because some of the proofs were unrigorous (such as (some of the moment-generating function stuff) and ___ (TO UPDATE LATER).
I'm taking off one star though, because some of the proofs were unrigorous (such as (some of the moment-generating function stuff) and ___ (TO UPDATE LATER).
January 17, 2019
This was the assigned textbook for my probability course. It's a great read with a lot of quality examples in the text. It didn't feel like it was jumping to conclusions, and it builds a good theoretical background for the conclusions it draws.
January 28, 2025
Great book and a must read for anyone beginning their journey in statistics. It has a ton of exercises and examples. Though it is a short book, it provides a thorough introduction to probability without being overwhelming.
August 27, 2018
Stanford 2018 summer class. CS109: Introduction to Probability for Computer Science.
October 25, 2018
Every time reading it.
Always new.
Always new.
July 2, 2020
The best book on probability theory
Want to read
October 24, 2020nice
May 10, 2021
A good introduction to probability. This textbook would be a great resource for individuals (even if you are not a Mathematics student) to learn more about the basics of the topic.
October 2, 2021
wide range of topics. the author ' explanation is usually not very rigorous or understandable. it still do a good job for what it intends to do.
November 28, 2021
If you are Statistical student , you must read this book!
August 8, 2025
ისევ ჯაფარიძე დაიწვას შუალედურის ქულოს გამო და FINALURI VINC GAMISWORA MADLOBA 30/30-STVIS. ეს წიგნი მართლა გადავღეჭე, მომეწონა.
April 3, 2024
didn’t finish; I’ll come back when I know more
July 26, 2024
I used this book as a first course in probability, which isn't that surprising given the title of this book. I'm not a math student and I used this book for self-study.
Probability Theory is hardcore compared to the classes you take before it. Having a gentle introduction to the field would be very nice. This definitely isn't a gentle introduction! The author didn't teach the topics in manageable chunks. The chapters are ordered in a way where all the exercises are put at the end of the chapter and it isn't exactly clear which exercises belong to which paragraph. This is a huge problem considering some chapters last 80+ pages! You have to digest 80 pages of information before having a chance to actually test yourself. This can be very tough on the mind.
The good thing is that there are a lot of exercises to test yourself on, so you can use a few exercises as dummy exercises to get into it. I didn't do all the exercises and will definitely come back from time to time to do a few more every few days, as learning takes time, and trust me, in probability theory, you'll need it!
Probability Theory is hardcore compared to the classes you take before it. Having a gentle introduction to the field would be very nice. This definitely isn't a gentle introduction! The author didn't teach the topics in manageable chunks. The chapters are ordered in a way where all the exercises are put at the end of the chapter and it isn't exactly clear which exercises belong to which paragraph. This is a huge problem considering some chapters last 80+ pages! You have to digest 80 pages of information before having a chance to actually test yourself. This can be very tough on the mind.
The good thing is that there are a lot of exercises to test yourself on, so you can use a few exercises as dummy exercises to get into it. I didn't do all the exercises and will definitely come back from time to time to do a few more every few days, as learning takes time, and trust me, in probability theory, you'll need it!
March 21, 2015
An introductory text in probability. As far as probability goes for most people, this book will be useful. Indeed, this is the recommended textbook to study for the first actuarial examination. The book is rich in examples and exercises that, as usual with probability, make you feel like playing games.
On the personal side, I regard this book high because it was the one used as the base to evaluate access to the graduate program in Statistics at the University of São Paulo.
On the personal side, I regard this book high because it was the one used as the base to evaluate access to the graduate program in Statistics at the University of São Paulo.
Want to read
July 9, 2009Good read. Very well written and clear thus far...
Read
October 6, 2009probability
Displaying 1 - 30 of 38 reviews