Probability theory arose originally in connection with games of chance and then for a long time it was used primarily to investigate the credibility of testimony of witnesses in the “ethical” sciences. Nevertheless, probability has become a very powerful mathematical tool in understanding those aspects of the world that cannot be described by deterministic laws. Probability has succeeded in ?nding strict determinate relationships where chance seemed to reign and so terming them “laws of chance” combining such contrasting – tions in the nomenclature appears to be quite justi?ed. This introductory chapter discusses such notions as determinism, chaos and randomness, p- dictibility and unpredictibility, some initial approaches to formalizing r- domness and it surveys certain problems that can be solved by probability theory. This will perhaps give one an idea to what extent the theory can – swer questions arising in speci?c random occurrences and the character of the answers provided by the theory. 1. 1 The Nature of Randomness The phrase “by chance” has no single meaning in ordinary language. For instance, it may mean unpremeditated, nonobligatory, unexpected, and so on. Its opposite sense is simpler: “not by chance” signi?es obliged to or bound to (happen). In philosophy, necessity counteracts randomness. Necessity signi?es conforming to law – it can be expressed by an exact law. The basic laws of mechanics, physics and astronomy can be formulated in terms of precise quantitativerelationswhichmustholdwithironcladnecessity.
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Part I. Probability. Basic Notions, Structure, Methods: Introduction; The Probability Space; Independence; General Theory of Stochastic Processes and Random Functions; Limit Theorems.- Part II. Markov Processes and Probability Applications in Analysis: Markov Processes; Probabilistic Representations of Solutions of Partial Differential Equations; Wiener Process and the Solution of Equations Involving the Laplace Operator.- Part III. Practical Probability Applications: Statistical Methods; Controlled Stochastic Processes; Information; Filtering.
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