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Bayes' theorem is a result in [[probability theory]] that shows how to invert a [[conditional probability]]. An extended form of the theorem provides a rigorous mathematical way of learning from uncertain data. This latter form is the basis of [[Bayesian statistics]].

'''Bayes' theorem''' is a result in [[probability theory]] that shows how to invert a [[conditional probability]]. An extended form of the theorem provides a rigorous mathematical way of learning from uncertain data. This latter form is the basis of [[Bayesian statistics]].

Let A and B be two uncertain events. We denote the probability of A by p(A) and probability of B by p(B). We denote the event that A and B are both true by AB, and the probability of AB by p(AB). In general, the fact that A has already occurred will affect the chances that B will occur, and vice versa. We denote the probability that B will occur, given that A has occurred, by p(B|A). We call this the probability of B, conditional on A. The event AB can occur in two ways. A can occur first, then B occurs, or B occurs first and then A occurs. The rules of probability require that the probability of two independent events both occurring is the product of the probabilities. Thus  

Let A and B be two uncertain events. We denote the probability of A by p(A) and probability of B by p(B). We denote the event that A and B are both true by AB, and the probability of AB by p(AB). In general, the fact that A has already occurred will affect the chances that B will occur, and vice versa. We denote the probability that B will occur, given that A has occurred, by p(B|A). We call this the probability of B, conditional on A. The event AB can occur in two ways. A can occur first, then B occurs, or B occurs first and then A occurs. The rules of probability require that the probability of two independent events both occurring is the product of the probabilities. Thus