Bayes' Theorem:

Bayes' Theorem:

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 A_i = 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?

.02*.85/(.02*.85+.98*.10)

= .017/(.017+ .098)

= .148