I've been dealing with statistics for a few years now. Up to now, for the probability part I've been referring to my old university book (my edition is even older, by the way), and of course the Internet. Since most of my statistics books are more recent (such as for example Bayesian Data Analysis by Andrew Gelman et al.), I was thinking about buying a more recent book for probability too. I am not looking for a rigorous, measure-theoretic book, thus this answer doesn't apply, but it doesn't have to shy away from all math either.

I've done a bit of research, and I found three books which sound interesting to me. I'm not restricting the choice to these three books: if you have a different suggestion, I'm all ears. The list is here only to establish some context, give an idea of the level of complexity I'm looking for, and to show that I've done a bit of homework :) I'm not coming here completely empty-handed. Feel free to suggest books outside of it, though.

Introduction to Probability, 2nd Edition, by Dimitri Bertsekas, John Tsitsiklis, 2008: the level looks simple enough that I can follow it easily (maybe even a bit too simple). I like the part on Bayesian inference: even though I have more advanced reference textbooks for that, something delving with the basics is always useful for a self-taught practitioner like me.

A First Course in Probability, Sheldon M Ross, 9th edition, 2013. This is cheap, which doesn't hurt :) the level of treatment again seems the right one, even if the Amazon preview is quite limited so I cannot be sure. Maybe a bit too many examples w.r.t. theorems/propositions, but I cannot really say.

Introduction to Probability (Chapman & Hall/CRC Texts in Statistical Science), by Joseph K. Blitzstein and Jessica Hwang, 2014. I like the fact that there's a chapter on MCMC: also, for what I've been able to see, the part on the limit theorems seems nice, focusing on intuition and applications, as well as on proofs.

As stated before, titles outside the list are welcome.

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    $\begingroup$ The last book's author has his course that is available freely online: projects.iq.harvard.edu/stat110/home , I'd highly recommend it. $\endgroup$ – Tim Nov 30 '16 at 13:58
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    $\begingroup$ The probability books dont have to be recent. Unless you're into research in the field, nothing has changed in past 50 years. $\endgroup$ – Aksakal Nov 30 '16 at 14:24
  • $\begingroup$ I remember liking your "old university book" (Papoulis) a lot when I was in school, and I still refer to my half-century-old copies of Feller Volumes I and II for probability theory. Don't know that I would consider MCMC to be probability theory per se. So I agree with @Aksakal. $\endgroup$ – EdM Nov 30 '16 at 15:04
  • $\begingroup$ @Aksakal, hmmm, I only partially agree. Sure, I'm not a researcher, but even if there were no new results, there are different applications now. For example, there's more focus on Bayesian inference. Notation used by Papoulis on the Kalman filter is different from what you see around now. Thus if I read a paper/presentation and I want to review some property/result the paper gives for granted, I find it a little bit harder to do with an older book than a new one. Or maybe it's the writing style/graphics that have changed? I don't know, but they felt a little bit easier to read for me. $\endgroup$ – DeltaIV Nov 30 '16 at 15:17
  • $\begingroup$ @DeltaIV, are you sure you mean probability theory and not statistics?Which part of Feller's book is too old for you these days? $\endgroup$ – Aksakal Nov 30 '16 at 15:21

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