Bayes' Theorem

An abstract definition of probability can be given by considering a set S, called the sample space, and possible subsets A, B,. .. , the interpretation of which is left open. The probability P is a real-valued function defined by the following axioms due to Kolmogorov [9]: From this definition and using the fact that A ∩ B and B ∩ A are the same, one obtains Bayes' theorem, From the three axioms of probability and the definition of conditional probability, one obtains the law of total probability,

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