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Bayesian Posterior Probability Calculator

Work out bayesian posterior probability instantly with clear inputs, formula shown and shareable results.

About Bayesian Posterior Probability Calculator

Bayes' theorem updates a prior with new evidence: posterior = prior × sensitivity / P(positive). When the prior is low, even a very accurate test yields many false positives, so the posterior stays surprisingly small.

Formula

Bayes' theorem

P(D|+) = P(D)·P(+|D) / [P(D)·P(+|D) + P(¬D)·P(+|¬D)]

Worked example

With the default values below, Bayesian Posterior Probability Calculator returns Posterior probability given a positive result of 16.666667. Change any input above and every figure here is recalculated in your browser.

Inputs used

Prior probabilityaccepts 0–100 %
1 %
Test sensitivity P(positive | condition)accepts 0.01–100 %
99 %
Test specificity P(negative | no condition)accepts 0.01–100 %
95 %

Results

Posterior probability given a positive result
16.666667 %
As a proportion
0.1666666667
Overall chance of a positive result
0.0594000000
Posterior given a negative result
0.0001063151

Step by step

  1. 1Values used
  2. 2Bayes' theorem
  3. 3Posterior probability given a positive result= 16.666667 %
  4. 4As a proportion= 0.1666666667
  5. 5Overall chance of a positive result= 0.0594000000
  6. 6Posterior given a negative result= 0.0001063151

Assumptions and limits

  • The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to 10 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For a 1% prior with 99% sensitivity and 95% specificity?

True positives are 0.0099 and false positives 0.0495, so the posterior is only 16.67% despite the accurate test.

Why is this counterintuitive?

Because the healthy group is so much larger that its 5% false-positive rate produces more positives than the sick group does.

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