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