# Probability

• Peter Schuster
Chapter
Part of the Springer Series in Synergetics book series (SSSYN)

## Abstract

Probabilistic thinking originated historically when people began to analyze the chances of success in gambling, and its mathematical foundations were laid down together with the development of statistics in the seventeenth century. Since the beginning of the twentieth century statistics has been an indispensable tool for bridging the gap between molecular motions and macroscopic observations. The classical notion of probability is based on counting and dealing with finite numbers of observations. Extrapolation to limiting values for hypothetical infinite numbers of observations is the basis of the frequentist interpretation, while more recently a subjective approach derived from the early works of Bayes has become useful for modeling and analyzing complex biological systems. The Bayesian interpretation of probability accounts explicitly for the incomplete but improvable knowledge of the experimenter. In the twentieth century, set theory became the ultimate basis of mathematics, thus constituting also the foundation of current probability theory, based on Kolmogorov’s axiomatization of 1933. The modern approach allows one to handle and compare finite, countably infinite, and also uncountable sets, the most important class, which underlie the proper consideration of continuous variables in set theory. In order to define probabilities for uncountable sets such as subsets of real numbers, we define Borel fields, families of subsets of sample space. The notion of random variables is central to the analysis of probabilities and applications to problem solving. Random variables are elements of discrete and countable or continuous and uncountable probability spaces. They are conventionally characterized by their distributions.

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