Bayes’ theorem, named after 18^{th} century (1763) British mathematician Thomas Bayes, is a mathematical formula for determining conditional probability. In the discussion of conditional probability we indicated that revising probability when new information is obtained is an important phase of probability analysis. Often, we begin our analysis with initial or prior probability estimates for specific events of interest. Then, from sources such as a sample, a special report, a product test, etc we obtain some additional information about the events. Given this new information, we update the prior probability values by calculating revised probabilities, referred to as *posterior probabilities*.

The steps involved in this probability revision process are depicted in the digram below:

#### Theorem:

An event A can occur only if one of the mutually exclusive and exhaustive set of events *B1, B2,… ,Bn* occurs. Suppose that the unconditional probabilities

And the conditional probabilities

are known. Then the conditional probability of a specified event *Bi*, when A is stated to have actually occurred, is given by

This is known as Bayes’ Theorem.

#### Proof:

An event A can happen in mutually exclusive ways, B1 A, B2A,… Bn A, i.e. either when has occurred, or. So by the theorem of total probability

Again,

Since the events *ABi* and *BiA* are equivalent, their probabilities are also equal.

Hence

So that

Substituting for *P(A)* from above, the theorem is proved.

Equation (1) is also known as “Bayes” formula for calculating probabilities of hypothesis. Because *B1, B2,…Bn* may be considered as hypothesis which account for the occurrence of A. The probabilities *P(B1),P(B2 ),…P(Bn)* are called ‘a prior’ probabilities of the hypothesis.

While are known as a‘a posteriori’ probabilities of the same hypothesis.

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