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These areas consider the use of numerical information to quantify observations about events. The tools and development are clearly mathematical; these areas overlap with analysis in particular. On the other hand, the use of the ideas developed here is primarily not mathematical areas so much as in the applications to the sciences.

• 60: 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.
• 62: Statistics is the science of obtaining, synthesizing, predicting, and drawing inferences from data. Elementary calculations of mean and standard variation suffice to summarize a large, finite, normally-distributed dataset; the field of Statistics exists since data are not usually so nicely given. If we do not know all the elements of the dataset, we must discuss sampling and experimental design; if the data are not normal we must use other parameters to summarize them, or resort to nonparametric methods; if multiple data are involved, we study the measures of interaction among the variables. Other topics include the study of time-dependent data, and the foundations necessary to avoid ambiguity or paradox. Computational methods (e.g. for curve-fitting) are of particular importance in applications to the sciences and engineering as well as financial and actuarial work.

Fields which contribute significantly to the development of these include 28: Measure and Integration; 05: Combinatorics; and 65: Numerical Analysis. Fields which make significant use of these include 85: Statistical mechanics; 90: Economics and operations research; and 92: Biology and behavioural sciences;

You might want to continue the tour with a trip through computer and information sciences.