Bayes theorem is a formula for revising a priori probabilities after receiving new information. The revised probabilities are called posterior probabilities. For example, consider the probability that you will develop a specific cancer in the next year. An estimate of this probability based on general population data would be a prior estimate; a revised (posterior) estimate would be based on both on the population data and the results of a specific test for cancer.
The formula for Bayes Theorem is as follows:
The best way to understand the terms is to look at an example. Consider a screening test for intestinal tumors. Let Ai = A1 = the event "tumor present", "B" the event "screening test positive" and "A2" the event "tumor not present" with no further A's.
If you have a tumor, the screening test has an 85% chance of catching it -- P(B|A1) = .85. However, it also has a 10% chance of falsely indicating "tumor present" when there is no tumor P(B|A2) = .10. The probability of a person having a tumor is .02 P(A1) = .02.
If the screening test is positive, what is the probability that you have a tumor?
= .017/(.017+ .098)