More than one combinator(ial?)ist has asked me to recommend a good book to learn probability from, and I never know what to say; the probability theory that I use in my research up was mostly learned piecemeal. (The stuff I learned in grad school from reading Chung and Feller hasn't been as useful, and I didn't especially enjoy those books.) Any suggestions?
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$\begingroup$ Nelson, "Radically Elementary Probability Theory" is a completely combinatorial account of probability theory. By using a very simple version of non-standard analysis, he can prove everything as a result on "finite" probability spaces. $\endgroup$– Michael GreineckerCommented Mar 29, 2012 at 19:10
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2$\begingroup$ What do you find particularly useful that is not in Feller? $\endgroup$– Igor RivinCommented Mar 29, 2012 at 19:29
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3$\begingroup$ This should probably be a wiki, as there are potentially multiple correct answers. $\endgroup$– Ori Gurel-GurevichCommented Mar 29, 2012 at 20:26
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2$\begingroup$ @Igor: I use a little bit of discrete potential theory (harmonic functions and Green's functions on graphs), Markov chain theory (especially the idea of coupling), and the stat mech formalism. I lean heavily on exact calculations via generating functions. I almost never need sigma-algebras or martingale technology, even though these were stressed in my graduate training. $\endgroup$– James ProppCommented Mar 29, 2012 at 21:35
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2$\begingroup$ The modern Feller is Grimmett and Stirzaker. I have personally never had any use for Feller as I find it dated and too much like a monograph. G&S, on the other hand, benefits from a very nice assortment of problems and solutions (in an accompanying volume). $\endgroup$– Steve HuntsmanCommented Mar 29, 2012 at 22:38
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The Probabilistic Method by Noga Alon and Joel Spencer!
Not a probability textbook per se ---Feller or whatever for that--- but sufficiently self-contained that one can learn the tools as one sees them applied -- to combinatorics!
answered Mar 29, 2012 at 19:39