Bayes' Theorem Calculator

About this calculator

This calculator applies Bayes' theorem, P(A|B) = P(B|A)·P(A) / P(B), to update a prior probability P(A) into a posterior probability P(A|B) once new evidence B is observed — using a test's sensitivity (true positive rate) and false-positive rate to work out how likely something is to actually be true given a positive result. It's the mathematical backbone of Bayesian inference: reasoning that formally combines what you believed beforehand with what new data tells you.

Doctors and epidemiologists use it to interpret medical test results correctly — a positive result for a rare disease can still mean a low actual probability of having it if the false-positive rate is high relative to the disease's prevalence. Statisticians, data scientists building spam filters and machine-learning classifiers, and forensic analysts weighing evidence all use this same reasoning; students learning probability use this calculator to build intuition for why “positive” doesn't always mean “likely.”