Bayes' Theorem: Updating a Probability with New Evidence
Bayes' Theorem describes how to update a prior probability belief once you observe new evidence:
P(A | B) = (P(B | A) · P(A))/P(B)
P(A | B) is the posterior — the updated probability of A once you know B happened. P(B | A) is the likelihood: how probable the evidence is if A is true. P(A) is the prior — what you believed before seeing any evidence. P(B) is the marginal probability of the evidence across every possibility, expanded with the Law of Total Probability:
P(B) = P(B | A) · P(A) + P(B | A^c) · (1 - P(A))
The single most important intuition here: the prior matters as much as the test itself. A highly accurate test applied to a rare condition still produces mostly false alarms, because there are so many more people without the condition than with it. That's exactly what this calculator is built to show.