Grimmett is probably a better bet to start with since it doesn't expect quite as much prior math knowledge and covers a lot more topics. The Williams book is shorter, denser and doesn't cover much in the way of applications, but gives a really good theoretical underpinning of how probability theorists think about probability (ie in terms of Lebesgue measures and Sigma algebras).
edit: There is also a companion book to Probability and Random Processes called One Thousand Exercises in Probability which contains an interesting selections of problems and solutions that will let you apply the theories taught in the main book.
Thanks, the first book (along with the exercises) seems like a great approach. Just peeked at the contents online, and it's pretty close to what I'm looking for.
Williams is probably better for mathematicians coming at probability with an already solid mathematical understanding rather than practitioners who want to try to understand the underlying theory.
- Probability and Random Processes by Grimmett
- Probability with Martingales by David Williams
Grimmett is probably a better bet to start with since it doesn't expect quite as much prior math knowledge and covers a lot more topics. The Williams book is shorter, denser and doesn't cover much in the way of applications, but gives a really good theoretical underpinning of how probability theorists think about probability (ie in terms of Lebesgue measures and Sigma algebras).
edit: There is also a companion book to Probability and Random Processes called One Thousand Exercises in Probability which contains an interesting selections of problems and solutions that will let you apply the theories taught in the main book.