Probability theory - Probabilitytheory Probability theory...

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Probability theory Probability theory is the branch of mathematics concerned with analysis of random phenomena. [1] The central objects of probability theory are random variables , stochastic processes , and events : mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion. Although an individual coin toss or the roll of a die is a random event, if repeated many times the sequence of random events will exhibit certain statistical patterns, which can be studied and predicted. Two representative mathematical results describing such patterns are the law of large numbers and the central limit theorem . As a mathematical foundation for statistics , probability theory is essential to many human activities that involve quantitative analysis of large sets of data. Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics . A great discovery of twentieth century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics . History The mathematical theory of probability has its roots in attempts to analyze games of chance by Gerolamo Cardano in the sixteenth century, and by Pierre de Fermat and Blaise Pascal in the seventeenth century (for example the " problem of points "). Christiaan Huygens published a book on the subject in 1657. [2] Initially, probability theory mainly considered discrete events, and its methods were mainly combinatorial . Eventually, analytical considerations compelled the incorporation of continuous variables into the theory. This culminated in modern probability theory, the foundations of which were laid by Andrey Nikolaevich Kolmogorov . Kolmogorov combined the notion of sample space , introduced by Richard von Mises , and measure theory and presented his axiom system for probability theory in 1933. Fairly quickly this became the undisputed axiomatic basis for modern probability theory. [3] [ edit ] Treatment Most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately. The more mathematically advanced measure theory based treatment of probability covers both the discrete, the continuous, any mix of these two and more. [ edit ] Discrete probability distributions Main article: Discrete probability distribution 1
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Discrete probability theory deals with events that occur in countable sample spaces. Examples: Throwing dice , experiments with decks of cards , and random walk . Classical definition: Initially the probability of an event to occur was defined as number of cases favorable for the event, over the number of total outcomes possible in an equiprobable sample space. For example, if the event is "occurrence of an even number when a
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Probability theory - Probabilitytheory Probability theory...

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