Bayes’ Theorem Calculator - Update a Probability with New Evidence
Bayes’ Theorem Calculator
How likely is it, now that you have the evidence?
Update a Belief
Start from how likely something was before the evidence and see how likely it is after.
Natural Frequencies
See how many people in a group of 10,000 are true and false positives, which makes the result easier to feel.
Tests and Screening
Shows why a positive result from an accurate test can still be probably wrong when the condition is rare.
Private
Nothing you enter is stored.
Why Base Rates Matter
Bayes’ theorem says how to revise a probability when you learn something new. Suppose 1 percent of people have a condition and a test catches 90 percent of cases but also gives a false alarm for 5 percent of healthy people. In 10,000 people, 100 have the condition and 90 of them test positive, while 9,900 are healthy and 495 of them test positive too. A positive result is therefore a true positive only 90 out of 585 times, about 15 percent, far lower than the 90 percent many people expect.
The reason is the base rate: when the condition is rare, even a small false-positive rate produces many more false alarms than real cases. The same reasoning applies to spam filters, security alerts, forensic matches and any screening. This calculator shows the probability after a positive and after a negative result, plus the expected counts, so you can see which side the evidence pushes the odds.
Key Takeaways
- Base rates dominate: Rare conditions give mostly false positives, even with accurate tests.
- Checked against exact fractions: Results match exact rational arithmetic.
- Related calculators: See also the Probability Calculator and the Confidence Interval Calculator.