Rating: 
Summary: Instructive and fun!
Review: An excellent introduction to probability and integration.Assuming a minimal background in elementary probability andundergraduate real analysis, this book uses humor, interesting examples, and fun exercises (solvable, though usually non-trivial) to introduce modern probability theory. Unlike so many math books where you need an instructor to highlight what's important, this book makes a point of letting the reader know where to concentrate his/her time by assigning star ratings to the theorems.
Rating: 
Summary: A Total Waste of Time
Review: As I have mentioned above, do not under any circumstances read (or even worse buy) this book. The layout is abolutely terrible and does the book no favours whatsoever. As indicated by previous reviews most of the proofs are completely glossed over with little or no mathematical rigour in sight. The appendices for the selected chapters are pretty much void of any use, and I can't see why the contents of these appendices could not have been included in the chapters in the first place (not that it would of made much of a difference). Hardly anything is explained with much clarity here and I don't know about anyone else out there but I would consider a one line example not much help at all. Everything right down to the way the index is presented is substandard in this book and the authors "attempts" at humour constituted the proverbial last straw for me (I find the assumption that all mathematics students like Star Trek somewhat insulting). In short this book is ABSOLUTE RUBBISH.
Rating:
Summary: A Great Introduction to Advanced Probability
Review: As the previous reviewers mentioned, this book is a concise and clear introduction to measure based probability. In contrast to some other reviewers, I like it a lot. The proofs ARE clear, the appendices ARE well placed and the order of presentation MAKES a lot of sense. One caveat is in order: since this book is EXTREMELY concise, one may have to think about the proofs and definitions for a while to digest it. The author does not chew up the material for you.
One more thing: This book is not just a rehashing of other textbooks on the topic. The order of presentation and proofs in this book is original (as far as I know) and if you find that you like it you are not likely to find it elsewhere.
A more detailed alternative that may serve as an excellent companion to this book is Probability and Measure by Ash.
Combine the book's originality of presentation with its clarity and its low price and the result is a great buy.
Rating:
Summary: Actually sucks knowledge from your brain
Review: I think perhaps that.... (The rest of this sentence is left to the reader). I know you were wondering why I used four dots instead of three but please trust me I will explain later. 'Think' is obviously used here instead of 'know'. One's intuition might suggest that know is a better word here but this is not the case! If you are feeling energetic then you could rearrange the letters in this review to produce a short collection of poetry.If you can read this review then by all means read this book. If you want to learn the theory of probability then don't. This book will confuse and baffle and ruin any knowledge you have. Maths is NOT hard, and it is not clever to make it seem hard. When people are writing books such as Kolmogorov's Intro. to Analysis and Oksendal's SDEs then no-one should be writing books like this.
Rating:
Summary: Excellent introduction
Review: I think this is the best introduction to modern probability and the theory of martingales. There is no unnecessary details, proofs are clear, exercises are interesting and challenging... After reading this book, one may read "Diffusions, markov processes and martingales"
Rating:
Summary: who said that probability is hard?
Review: Start reading this book, and soon you'll discover that you can learn more than you ever thought you could. The usual "Williams" style, but for the layman [almost]. Humor (humour) makes the book friendly. The section on a discrete-time version of the Black-Scholes formula can be improved... Could the author perhaps tell the story when the interest rate takes more than 2 values? In short, here is a book to cure those who think that (Omega, F, P) is redundant abstract nonsense. I also learned what "Mabinogion" stands for, but have not yet explained the seemingly "greek" sounding etymology. Maybe my Welsh needs to be improved :-) Takis Konstantopoulos
Rating:
Summary: who said that probability is hard?
Review: Start reading this book, and soon you'll discover that you can learn more than you ever thought you could. The usual "Williams" style, but for the layman [almost]. Humor (humour) makes the book friendly. The section on a discrete-time version of the Black-Scholes formula can be improved... Could the author perhaps tell the story when the interest rate takes more than 2 values? In short, here is a book to cure those who think that (Omega, F, P) is redundant abstract nonsense. I also learned what "Mabinogion" stands for, but have not yet explained the seemingly "greek" sounding etymology. Maybe my Welsh needs to be improved :-) Takis Konstantopoulos
Rating:
Summary: eccentric, but wonderful
Review: The reviewer who rated this a single star gives a decent imitation of Williams' prose style. What he doesn't mention is Williams' infectious enthusiasm for probability, the beautiful proofs, and the conciseness of this book. You should, of course, read Feller vol. 1 first, but this would be my next choice. I'd never really appreciated rigorous probability before reading this book. He shows that it's not all technicalities.
Rating: 
Summary: It's Great, But You'll Have To Think
Review: This is a really solid book that covers the basic theory of Martingales in discrete time well. If you're serious about learning probability for real, this is as good a place to start as any. Just make sure you have a pencil and paper with you when you're reading it - David Williams expects you to do some of the work for yourself. And quite right too.
Rating:
Summary: It's Great, But You'll Have To Think
Review: This is a really solid book that covers the basic theory of Martingales in discrete time well. If you're serious about learning probability for real, this is as good a place to start as any. Just make sure you have a pencil and paper with you when you're reading it - David Williams expects you to do some of the work for yourself. And quite right too.
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