Bayes' theoremBayesTheorem2001-12-03 23:52:052009-04-02 15:01:41TheoremstatisticsBayes\usepackage{amssymb}
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Let $(A_n)$ be a sequence of mutually exclusive events whose \PMlinkname{union}{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).
\begin{thebibliography}{3}
\bibitem{Milton} Milton, J.S., Arnold, Jesse C., \textsl{Introduction to Probability and Statistics: Principles and Applications for Engineering and the Computing Sciences}, McGraw Hill, 1995.
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