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Calculation Machine

Bayes Theorem Calculator

Turn a test result into a real posterior probability.

Reviewed by James Whitfield · How we check our maths

Your values

ⓘ Live results

Results
P(condition | positive test)
16.102%
P(condition | negative test)
0.0531%
Probability of a positive test
5.9%
Share of positives that are false
83.9%
Specificity
95%
Positive likelihood ratio
19

Quick answer

Bayes' theorem updates a prior with new evidence: P(A|B) = P(B|A)·P(A) ÷ P(B), which is why a positive test on a rare condition is often still a false alarm.

The formula

P(A|B) = P(B|A)·P(A) ÷ P(B)

Worked example

Worked example

Using the values the calculator loads with.

Inputs

  • Prior probability (base rate)1 %
  • Sensitivity — true positive rate95 %
  • False positive rate5 %

Result

  • P(condition | positive test)16.102%
  • P(condition | negative test)0.0531%
  • Probability of a positive test5.9%
  • Share of positives that are false83.9%
  • Specificity95%
  • Positive likelihood ratio19

How to use the Bayes Theorem Calculator

  1. 1Enter how common the condition is in the population you are testing.
  2. 2Enter the test's sensitivity and false positive rate.
  3. 3Read the posterior — the real chance the result is true.

Frequently asked questions

Why is a 95% accurate test often wrong?

With a 1% base rate, the far larger healthy group generates more false positives than the small sick group generates true ones.

How do I improve the posterior?

Test a higher-risk group (raising the prior) or retest — a second independent positive multiplies the likelihood ratio.

About this calculator