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What are the topics one should know before delving into probability theory? (Please recommend any books you know on those topics too.) I think there is set theory, but set theory is a large topic in itself. Does probability need only a little bit of set theory? Also, there is combinatorics, but again combinatorics is big in itself. And is it a good idea to know all topics such as set theory and combinatorics to fully understand probability theory? Or is it enough just to read those topics on the fly?

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Dependending on how deeply you want to explore the field, you will need more or less.

If you want a basic introduction then some basic set theory (what is a set and elementary set operations), combinatorics (knowing different ways of counting, inclusion-exclusion principle) and calculus (knowing derivatives and integrals). This could get you through a basic text in probability.

If you want more serious stuff, I would study measure theory (which serves as the foundation of probability through Kolmogorov's axioms), a thorough knowledge of analysis that goes beyond just knowing calculus, maybe even some functional analysis, combinatorics and generally some discrete mathematics (like working with difference equations).

This will allow you to follow a solid introductory course on probability. After that, it depends a lot on what related branches you want to explore. If you want to study Markov chains, a good knowledge of linear algebra is a must. If you want to delve deeper into statistics (like hypothesis testing and such) more analysis will do you good, etc...

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Great answer. To the OP, it might be useful to note that you don't necessarily need to know measure theory before delving into probability theory though. There are actually many books on probability that introduce abstract measure theory from the ground up as well, which might not be worse than the more formal method with Lebesgue measure. –  user1736 Jan 13 '11 at 21:53
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Combinatorics is a very large subject but one set of counting problems that is very helpful for discrete probability is counting the number of inequivalent ways to place balls in boxes. Fundamental cases include when the balls are considered distinguishable or indistinguishable and the boxes are considered distinguishable and indistinguishable. Fred Roberts' book Applied Combinatorics does a nice job with this material.

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It depends which kind of probability theory you're interested in. An introductory course on probability theory can either dwell on discrete probability or continuous probability.

Discrete probability, which deals with discrete events (e.g. the probability that if you throw a dice it comes up $6$ ten times in a row), only really needs elementary combinatorics. From set theory you need to know the definitions of basic concepts, and from combinatorics you need to know the likes of the binomial coefficient and its properties.

A little more is needed to understand Poisson random variables, namely Stirling's approximation, which is a topic you don't really learn anywhere; this is why these courses often just give the definition, which requires you to know the Taylor expansion of $e^x$. But this topic in its entirety is not necessarily covered.

Continuous probability deals with things like the normal distribution and the central limit theorem - distributions which may take "continuous" values (e.g. every real value rather than only integral values). Sometimes it is given as an addendum to a discrete probability course. To understand continuous probability you will need to know basic calculus (the kind you get from a first course, and then some).

Introductory courses don't usually cover multivariate Gaussians, but these require some linear algebra.

Summarizing, you will need to be confident about some fairly basic topics. Besides some familiarity with basic concepts, it's also best to have some "mathematical maturity", although not too much of it is actually needed in an introductory course.

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For elementary, probability theory you can look into these two books:

  • A First Course in Probability by Sheldon Ross

  • An Introduction to Probability Theory and Its Applications, by W.Feller.

  • Introduction to Probability and Measure by K. R. Parthasarathy.

Both books provide very good introduction to the subject. Moreover, it would be nice if you know some basic calculus and set theory because you may need them when you study about Distribution functions of Various Random variables.

The last book which i have added is a really nice book. It's available in Indian edition but i am not sure about it's sales in foreign.

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Feller is definitely a classic. –  Raskolnikov Jan 13 '11 at 15:09
    
@Raskolnikov: Thanks. Yes i forgot to add measure theory, and i am happy you added it. –  anonymous Jan 13 '11 at 15:10
    
Thanks for the book recommendations. As I understand Feller's book is in two volumes? –  Baha Jan 14 '11 at 2:31
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