Bayes’ theorem

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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.