Direct answer
Bayes’ theorem combines a prior probability with evidence likelihoods; when the prior event is rare, false positives among the much larger non-event group can keep the posterior modest.
Visual explanation
Base rates shape the positive-result denominator
What this calculation tells you
Bayesian updating asks how probability should change after specified evidence. It distinguishes the probability of evidence given a hypothesis from the probability of the hypothesis given the evidence.
Use a prior and likelihoods defined for the same population, evidence threshold, and conditions. Translate percentages into expected counts when the denominator feels unintuitive.
Where it is used
Quality inspection
Update defect probability from entered inspection performance.
Fraud review
Understand how rare-event prevalence affects alerts.
Research
Translate priors and likelihoods into a transparent posterior calculation.
Education
Expose inverse-probability errors using natural frequencies.
When this guide helps
- Interpreting a positive alert for a rare event.
- Comparing two plausible priors.
- Explaining why accuracy is not posterior probability.
- Auditing a multi-hypothesis update.
Start with the statistical question
Use a prior and likelihoods defined for the same population, evidence threshold, and conditions. Translate percentages into expected counts when the denominator feels unintuitive.
A grid of 1,000 people separates the prior-positive and prior-negative groups, then colors true and false positive evidence outcomes. The posterior is the true-positive share of all positive outcomes.
Worked example
With a 1% prior, 90% sensitivity, and 5% false-positive probability, 1,000 cases produce about 9 true positives and 49.5 false positives. The posterior after a positive is about 9/58.5=15.4%, not 90%.
Assumptions that carry the result
The prior must be relevant to the individual or population being analyzed, and likelihoods must be estimated under compatible definitions. Dependence between repeated pieces of evidence requires a joint model.
Interpret the result without overreaching
The calculation does not choose a prior, establish clinical meaning, validate evidence quality, or make a treatment or policy decision. Consequential use requires domain review.
- Confusing P(evidence|hypothesis) with P(hypothesis|evidence).
- Ignoring the base rate.
- Multiplying repeated likelihoods as if evidence were conditionally independent.
Worked case: rare-condition screen
Prevalence 1%, sensitivity 90% and false-positive rate 5% in the modeled population.
True-positive share = 0.01 × 0.90 = 0.009; false-positive share = 0.99 × 0.05 = 0.0495; posterior = 0.009/0.0585 = 15.38%.
About 15.4% of modeled positive screens are true cases under these inputs.
The result is not an individual diagnosis and depends strongly on population and test definitions.
Reproduce this worked caseOpen Bayes’ Theorem Calculator
Worked case: higher base rate
Use prevalence 10% with the same sensitivity and false-positive rate.
Posterior = 0.10 × 0.90/(0.09+0.90 × 0.05) = 0.09/0.135 = 66.67%.
The same test characteristics yield a much higher posterior in a higher-prevalence population.
Do not transport a posterior between populations without the base rate.
Reproduce this worked caseOpen Bayes’ Theorem Calculator
Bayes theorem: compare assumptions, not just answers
Verify sensitivity, specificity, prevalence, reference standard and population. Calculator arithmetic cannot provide clinical interpretation or treatment.
| Case | Calculation focus | Interpretation |
|---|---|---|
| Prevalence 1% | Same test | 15.38% |
| Prevalence 10% | Same test | 66.67% |
Bayes theorem: calculation checklist
- Events defined
- False-positive vs specificity correct
- Base rate population matched
- Denominator includes both positive paths
- No diagnosis
Practical questions
Frequently asked questions
Is the posterior the same as sensitivity?
No. Sensitivity conditions on the event; the posterior conditions on the observed evidence and also depends on the prior and false-positive rate.
Where does the prior come from?
It must come from a justified population, previous evidence, or explicit model; Bayes’ theorem does not select it automatically.
Can I update twice?
Yes when the sequential evidence model is justified. Simply multiplying likelihoods assumes a conditional relationship that may not hold.
Further reading
Authoritative sources
Use these primary and professional resources to check definitions, conventions, or requirements that may extend beyond this guide.