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# Bayes’ theorem

Let $(A_{n})$ be a sequence of mutually exclusive events whose union is the sample space and let $E$ be any event. All of the events have nonzero probability ($P(E)>0$ and $P(A_{n})>0$ for all $n$). Bayes’ Theorem states

$P(A_{j}|E)=\frac{P(A_{j})P(E|A_{j})}{\sum_{i}P(A_{i})P(E|A_{i})}$ |

for any $A_{j}\in(A_{n})$.

A simpler formulation is:

$P(A|B)=\frac{P(B|A)P(A)}{P(B)}$ |

For two events, $A$ and $B$ (also with nonzero probability).

# References

- 1 Milton, J.S., Arnold, Jesse C., Introduction to Probability and Statistics: Principles and Applications for Engineering and the Computing Sciences, McGraw Hill, 1995.

Keywords:

statistics, Bayes

Related:

ConditionalProbability

Synonym:

Bayes' Rule

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

60-00*no label found*62A01

*no label found*

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