I am looking for a good book to study probability. My advisor suggested the "Probability" by Leo Breiman. I am reading it now, it seems rather a dense book, so I would like to ask you guys advice on which book you guys often start with for Probability.

10$\begingroup$ It's hard to go wrong with the twovolume set by Grimmett and Stirzaker: vol. 1 is the main text and vol. 2 are the exercise solutions. I think of this as a modern counterpart to Feller. It focuses on problems and is quite accessible, but introduces all the basic ideas that any serious student will have to master. If you are doing mathematical finance you can't go wrong with Williams' Probability with Martingales. It includes a section on a discrete version of BlackScholes and will provide solid preparation for the real stuff. $\endgroup$– Steve HuntsmanCommented Apr 5, 2010 at 3:08

$\begingroup$ Note that Breiman uses the (at least what used to be) Russian convention of leftcontinuous distribution function $\endgroup$– Mark L. StoneCommented Jul 8, 2018 at 14:27

1$\begingroup$ @MarkL.Stone :o $\endgroup$– Michael HardyCommented Nov 19, 2023 at 17:51
14 Answers
I would definitely go for "Probability" by Jim Pitman. It is a very good book for learning Probability Theory, one of the best text books I have encountered in my studies.
Williams' book Probability with Martingales is short, thorough and fun, and will be a good introduction to the kind of probability theory used in finance.

5$\begingroup$ Can't upvote this enough. Williams teaches you how to think probabilistically without too much technical machinery, so this is a great book even if you have no interest in mathematical finance. $\endgroup$ Commented May 11, 2010 at 1:53
Rick Durrett's book "Probability: Theory and Examples" is a very readable introduction to measuretheoretic probability, and has plenty of examples and exercises. This is the second text that I learned probability theory out of, and I thought it was quite good (I used Breiman first, and didn't enjoy it very much). As a bonus, there is a free .pdf version of the 4th edition (which will be published in a few months) available on his website for the time being.
A recent text covering similar material (which I admit I haven't read that fully) which looked good on a quick reading was "Probability Theory: A Comprehensive Course" by Klenke. It has a very nice selection of modern topics.

3$\begingroup$ I'm surprised to hear Durrett's book called "very readable". While it may be very readable for you, I don't doubt that at all, for me it was quite hard to get anywhere. I think if you have a STRONG background in analysis and some exposure to concrete probability (like an upperdiv stat class or something) Durrett's book might be easier. $\endgroup$ Commented May 14, 2010 at 21:01
I really enjoyed this unfinished book by Rota and Baclawski on probability and random processes. It is extremely well written (like everything else by Rota) and covers a number of interesting topics like Markov processes, entropy and information, and Brownian motion.
Grimmett and Stirzaker is a very good book, but it's quite difficult for the average student. The students at Cambridge who trained on these books are some of the best in the world.
The best all around probability text for serious undergraduates right now is (forgive the pun) probably PROBABILITY AND STATISTICS, 3ed edition, by DeGroot And Shervish. Most undergraduate textbooks aren't comprehensive enough to cover important functions like the gamma and beta families of distributions. This one does and does it very carefully and rigorously without going too far into the theoretical side.
I second the suggestion of Hoel, Port and Stone from jamie above. It's terrific and mathematically perfect. I first learned probability from it and the wonderful lectures of Stefan Ralescu. My one complaint with it is that it's ridiculously expensive. That's why I suggested DeGroot and Sharevich instead. It covers the same material and more and at a better price. If you can borrow HPS, thoughby all means, please do so.
There is a very good book available for free from the Scuola Normale Superiore of Pisa: Introduction to Measure Theory and Probability.
Suggestions on probability books: I strongly recommend Billingsley's Probability and Measure, this book includes three parts: (1) measure theory (2) probability theory (3) stochastic process, this book mainly focuses on the first two parts. This book is very very nice ! Its ideas and proofs are beautiful and friendly, and mathematical rigorously.
Additionally, Rick Durrett's probability:theory and examples, this book is very popular and appointed as text book in many (top) universities, but it doesn't involve too much about measure theory, and for some proofs and examples given in the book, to be honest, I cannot understand clearly, maybe Rick thinks that's very easy. All in all, I don't feel happy and excited when I read this book.
Kai Lai Chung's A course in probability theory, also very popular, Chung is a very famous and excellent probabilist， I just read part of this book, feel that's very good ! Chung is Rick's advisor ~~ !
William's probability with martingale is also a good book, I just focus on martingale part of this book, you can have a look.
Probability Theory(Yuan Shih Chow and Henry Teicher) is also a popular book in academia. Doob is Chow's advisor, and Chow mainly studies martingale theory.

$\begingroup$ Varadhan's probability theory is not a dense book, that's mainly a collection of his lecture notes, maybe you like it. $\endgroup$– chengCommented Jul 8, 2018 at 14:18

$\begingroup$ If you want to study probability theory in the future, shiryaev's probabilty is a good reference, and it also involves time series theory and stochastic process. Two volumes ! $\endgroup$– chengCommented Jul 8, 2018 at 14:20
I learned from this book by Hoel, Port and Stone. I didn't find it very dense. I thought it had a lot of good examples. If you're getting bogged down in the first 4 chapters of Breiman's book, it might be a good introduction before you learn some of the deeper stuff.

$\begingroup$ HPS is much cheaper from amazon.co.uk $\endgroup$ Commented Apr 6, 2010 at 15:35
I agree with your prof on this one, Breiman is a good book and very readable within this field, though someone above said Durrett was very readable... I suppose it is a matter of taste to some degree. If you're looking for something that's less dense than Breiman, I would highly recommend avoiding Durrett.
A few years ago I wrote my own `fastmoving' introduction to probability theory which may help you to see the route ahead when reading a book like Williams's Probability With Martingales.
You'll find the PDF article in my blog post here.
I am going to shameless plug Erhan Cinlar's GTM volume Probability and Stochastics. It is maths oriented, but it doesn't assume too much (since it was based on a course taught in the Operations Research department in the engineering school), so the first thing it does is measure theory and starts from there. And it also has lots of nice exercises.
I am a bit sad that for the final published version the author went back and removed most (if not all) of the wry remarks on the history of mathematics.
I would suggest "Probability: A Graduate Course" by Alan Gut. It covers key topics which are taught in a typical probability theory course. In addition, (almost) all proofs are very detailed and accessible.
If you read French fluently, I suggest "de l'intégration aux probabilités" by Olivier Garet and Aline Kurtzmann, which builds the theory of probability in the framework of measure theory with number theoretic applications.
The fifth (2019) edition of the book "Probability: Theory and Examples" by Rick Durrett is unsurpassed in clarity (and wit!) by any textbook on probability theory of which I am aware, that treats the subject in its proper, measuretheoretic setting. The examples promised in the title are beautiful and illuminating! The discussion of conditional expectation of a random variable with respect to another is particularly praiseworthy! It contains an extensive discussion of illuminating examples. I can't imagine how it could be improved! This is a GREAT book that everyone interested in probability or statistics MUST own!
The same can be said for the encyclopedic "Foundations of Modern Probability," by O. Kallenberg (second edition, 2007). It is truly a monumental achievement! It will almost surely be the standard reference on the subject for decades to come! The entire mathematical community owes an huge measure of gratitude to Professor Kallenberg for the enormous amount of effort that he obviously spent writing this wonderful book! It's comparable to the famous threevolume classic "Linear Operators," by Dunford and Schwartz or the equally famous sixvolume classic, I.M. Gelfand's "Generalized Functions." Or perhaps even to the ninevolume (!) "Treatise on Analysis" by J. Dieudonne!