Bayes' Theorem Calculator

This calculator applies Bayes' theorem to find an updated, or posterior, probability after taking new evidence into account, the cornerstone of reasoning under uncertainty. Bayes' theorem tells you how to revise a probability when you learn something new, combining what you believed beforehand, the prior, with how strongly the new evidence points one way or the other. Its most famous and counter-intuitive application is medical testing: even an accurate test for a rare condition can produce a surprisingly low chance of actually having the condition after a positive result, because most positives come from the large healthy population. The same logic underlies spam filters, fault diagnosis, machine learning and everyday judgement. This tool makes the classic test scenario concrete. You enter the prior probability of having the condition, often the prevalence in the population, the test's sensitivity, the chance it correctly flags a true case, and the false positive rate, the chance it wrongly flags a healthy person. The calculator returns the posterior probability of genuinely having the condition given a positive test, along with the overall probability of a positive result and the inputs for reference. The results update as you type. Use it to understand test results properly, for statistics and probability study, or to see why screening rare conditions produces many false alarms. The key lesson the calculator drives home is that a positive result from a good test is far from a certainty when the condition is rare: with a one percent prevalence and a five percent false positive rate, even a 99 percent accurate test leaves only about a one in six chance of truly having the condition. Bayes' theorem turns this intuition into a precise number.

Bayes: P(A|B) = P(B|A)P(A) / [P(B|A)P(A) + P(B|not A)P(not A)]. With a rare condition, even an accurate test gives many false positives.