Bayes Theorem Calculator

Bayes theorem is the mathematical formula for updating a probability estimate when you receive new evidence. It underpins much of modern statistics, machine learning, medical diagnosis, and reasoning under uncertainty. The theorem connects three probabilities you usually know or can estimate to produce one you want to find out. The prior probability P(A) is how likely the event or condition is before you see the evidence. The likelihood P(B|A) is how likely the evidence would be if the event is true. The false positive rate P(B|not A) is how likely the evidence would appear even if the event is not true. Combining these three using Bayes theorem gives you the posterior probability P(A|B): the updated probability that A is true now that you have observed B. One of the most important and counterintuitive lessons from Bayes theorem is how much the base rate matters. If a condition is very rare, even a highly accurate test will produce mostly false positives, because the large pool of unaffected people swamps the small pool of truly affected people. With a prior of 1%, a sensitivity (true positive rate) of 90%, and a false positive rate of 10%, the posterior probability is only about 8.33%. This calculator accepts all three inputs as percentages or decimals and returns the posterior P(A|B), the total probability of observing B, and the likelihood ratio. It is useful for students, medical students learning diagnostic reasoning, engineers working on signal detection, and anyone applying Bayesian thinking to real-world problems.

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