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[QUICK DESCRIPTION]

This article gives an introductory description of probability theory and links to Tricki articles on methods of this field.

[GENERAL DISCUSSION]

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In mathematics and its applications it often makes sense to assign weights to elements of a set of interest. A weight can mean many things, for example, the statistical frequency of an observation of a random outcome, allocation of a resource or the degree of belief in an outcome. Then a typical problem is to compute the total weight of a subset of elements. Probability theory studies this and related problems, the ways one can assign weights to various sets and how these weights transform as the sets on which they are defined are transformed.

The concept that precisely defines the idea of assigning generalized weights to sets is that of a measure. A probability measure is a nonnegative measure with total measure $1$.

In probability theory [[w:Measurable Function| measurable functions]] are called [[w:Random variable|random variables]]. An integral against a probability measure is called an expectation.

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The main areas of study in probability theory are: 
* Construction: how to define and characterize probability measures,
* Transformations of probability measures,
* Convergence: 
** Given a sequence of probability measures or random variables, what are the possible ways for this sequence to converge? 
** In case of taking limits of measures, what are the possible limit measures? 
** How to characterize and check convergence? 
** Analysis of the speed of convergence,
** Asytmptotics of expectations,
* Computation of probabilites and expectations,
* Bounding probabilities and expectations,
* Optimizing probabilities and expectations,
* Min/Max problems involving expectations.

Probability theory also studies the structure of probability measures. In particular, the theory of stochastic processes, i.e., the study of measures on spaces of functions of a real or an integer variable, is very advanced.

This page contains links to more specific navigational pages devoted to probability.

[[Elementary probability front page]]

[[Stochastic processes front page]]

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