Bayes’ theorem

states

Let $(A_{n})$ be a sequence of mutually exclusive events whose union (http://planetmath.org/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.
Title Bayes’ theorem BayesTheorem 2013-03-22 12:02:13 2013-03-22 12:02:13 akrowne (2) akrowne (2) 11 akrowne (2) Theorem msc 60-00 msc 62A01 Bayes’ Rule ConditionalProbability