Write a short note on probability theory
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probability theoryis the branch ofmathematicsconcerned withprobability. Although there are several differentprobability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ofaxioms. Typically these axioms formalise probability in terms of aprobability space, which assigns ameasuretaking values between 0 and 1, termed theprobability measure, to a set of outcomes called thesample space. Any specified subset of these outcomes is called anevent.Central subjects in probability theory include discrete and continuousrandom variables,probability distributions, andstochastic processes, which provide mathematical abstractions ofnon-deterministicor uncertain processes or measuredquantitiesthat may eitherbe single occurrences or evolve over time in a random fashion.Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability theory describing such behaviour are thelaw of large numbersand thecentral limit theorem.As a mathematical foundation forstatistics, probability theory is essential to many human activities that involve quantitative analysis of data.[1]Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as instatistical mechanics. A great discovery of twentieth-centuryphysicswas the probabilistic nature of physical phenomena at atomic scales, described inquantum mechanics.[2]
probability theoryis the branch ofmathematicsconcerned withprobability. Although there are several differentprobability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ofaxioms. Typically these axioms formalise probability in terms of aprobability space, which assigns ameasuretaking values between 0 and 1, termed theprobability measure, to a set of outcomes called thesample space. Any specified subset of these outcomes is called anevent.Central subjects in probability theory include discrete and continuousrandom variables,probability distributions, andstochastic processes, which provide mathematical abstractions ofnon-deterministicor uncertain processes or measuredquantitiesthat may eitherbe single occurrences or evolve over time in a random fashion.Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability theory describing such behaviour are thelaw of large numbersand thecentral limit theorem.As a mathematical foundation forstatistics, probability theory is essential to many human activities that involve quantitative analysis of data.[1]Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as instatistical mechanics. A great discovery of twentieth-centuryphysicswas the probabilistic nature of physical phenomena at atomic scales, described inquantum mechanics.[2]
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