``A priori probability is the probability estimate prior to receiving new information.
Posterior probability is a revised probability that takes into account new available information. For example, let there be two urns, urn A having 5 black balls and 10 red balls and urn B having 10 black balls and 5 red balls. Now if an urn is selected at random, the probability that urn A is chosen is 0.5. This is the a priori probability. If we are given an additional piece of information that a ball was drawn at random from the selected urn, and that ball was black, what is the probability that the chosen urn is urn A? Posterior probability takes into account this additional information and revises the probability downward from 0.5 to 0.333 according to Bayes’ theorem, because a black ball is more probable from urn B than urn A.
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 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)