60: Probability theory and stochastic processes
Probability theory is simply enumerative combinatorial analysis when applied to finite sets; thus the techniques and results resemble those of discrete mathematics. The theory comes into its own when considering infinite sets of possible outcomes. This requires much measure theory (and a careful interpretation of results!) More analysis enters with the study of distribution functions, and limit theorems implying central tendencies. Applications to repeated transitions or transitions over time lead to Markov processes and stochastic processes. Probability concepts are applied across mathematics when considering random structures, and in particular lead to good algorithms in some settings even in pure mathematics.
Some material in probability (especially foundational questions) is really measure theory. The topic of randomly generating points on a sphere is included here but there is another page with general discussions of spheres. Probability questions given a finite sample space are usually "just" a lot of counting, and so are included with combinatorics.
For additional applications, See 11KXX, 62-XX, 90-XX, 92-XX, 93-XX, 94-XX. For numerical results, See 65U05
This is one of the larger areas in the Math Reviews database. (The subfield 60K25, Queueing theory, is among the largest of the five-digit subfields.)
Browse all (old) classifications for this area at the AMS.
Bosch, A. J.: "What is a random variable?" (Dutch) Nieuw Tijdschr. Wisk. 64 (1976/77), no. 5, 241--250. MR58#2932
Doob, J. L.: "What is a martingale?", Amer. Math. Monthly 78 (1971) 451--463. MR44#1094
Doob, J. L.: "What is a stochastic process?", Amer. Math. Monthly 49 (1942) 648--653. MR4,103b
Resnick, Sidney: "Adventures in stochastic processes", Birkhäuser Boston, Inc., Boston, MA, 1992. 626 pp. ISBN 0-8176-3591-2 MR93m:60004
Ocone, Daniel L.: "A guide to the stochastic calculus of variations", Stochastic analysis and related topics (Silivri, 1986), 1--79, Lecture Notes in Math., 1316; Springer, Berlin-New York, 1988. MR89h:60093
Wise, Gary L.; Hall, Eric B.: "Counterexamples in probability and real analysis", Oxford University Press, New York, 1993, ISBN 0-195-07068-2
Stoyanov, Jordan M., "Counterexamples in probability", Wiley, Chichester-New York, 1987, ISBN 0-471-91649-8
An online course guide for a course in Introductory Probability.
There is a node in the GAMS software tree for Simulation, stochastic modeling.
There is a newsgroup alt.sci.math.probability .