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Bayesian Calculator

Bayes' theorem updates a probability when new evidence arrives. It is used to interpret medical tests, spam filters and risk assessments.

How it works Report a problem

Bayesian CalculatorBayesian

Probabilistic Inference Intelligence

Initial prevalence (%)
True Positive Rate (%)
True Negative Rate (%)

About the Bayesian Calculator

Bayes' theorem updates a probability when new evidence arrives. It is used to interpret medical tests, spam filters and risk assessments.

How to use the Bayesian Calculator

  1. Enter the Prior P(A), the probability that the condition is present before testing.
  2. Enter the test's Sensitivity (true positive rate) and Specificity (true negative rate).
  3. Press Calculate to see the probability after a positive result.

Formula and method

P(A | positive) = (Sensitivity × Prior) ÷ (Sensitivity × Prior + (1 − Specificity) × (1 − Prior))

Worked example

A condition affects 1% of people. A test has 99% sensitivity and 95% specificity. After a positive result the probability of having the condition is only 16.7%, because false positives outnumber true positives when the condition is rare.

Tips

  • Enter percentages as decimals if the tool asks for values between 0 and 1.
  • Rare conditions produce many false positives even with accurate tests.

Frequently asked questions

Why is the result so much lower than the test accuracy?
Because the prior probability is small. Most positives come from the large healthy group.