Answer first
What this calculator tells you
Calculate the chance of a condition after a positive or a negative test from prevalence, sensitivity and false positive rate. See why a positive result on a good test can still mean a low chance of the condition when it is rare. Formula: P(condition | positive) = sensitivity × prevalence ÷ [sensitivity × prevalence + false positive rate × (1 − prevalence)]. At the worked-example inputs, the chance of the condition after a positive test is 15.4%. Holding every other input steady, moving prevalence (prior chance) from 0.500% to 1.5% moves the result from 8.3% to 21.5%.
Transparent method
The formula
See why a positive result on a good test can still mean a low chance of the condition when it is rare.
Worked example
Example inputs
How to interpret the result
Bayes' theorem updates a probability when new evidence arrives, and it shows how much the starting rate matters. With a condition affecting 1 percent of people, a test that is 90 percent sensitive and gives 5 percent false positives returns a positive result that means only about a 15 percent chance of the condition. Most positives come from the 99 percent who are healthy, since 5 percent of a large group beats 90 percent of a small one.
At the worked-example inputs the chance of the condition after a positive test is 15.4%. It rises with prevalence (prior chance) and sensitivity and falls as false positive rate increases.
These are standard formulas from introductory statistics. Real-world inference depends on assumptions, normality, independence, a genuinely random sample, that this calculator does not check for you.
Before you rely on it
What to check
Use the prevalence for the group being tested. A person with symptoms starts from a much higher prior than one tested at random, and the same test means more for them.
The common error
Where people go wrong with bayes' theorem calculator
Reading sensitivity as the chance of having the condition after a positive test. Those are different numbers, and mixing them up is one of the most common errors in medical testing.
Sensitivity evidence
How prevalence (prior chance) changes the chance of the condition after a positive test
Holding every other input at the worked-example value, moving prevalence (prior chance) from 0.500% to 1.5% moves the chance of the condition after a positive test from 8.3% to 21.5%: a spread of 13.2%, or 86% of the worked-example result.
| Prevalence (prior chance) | Chance of the condition after a positive test | Chance of a positive test overall | Chance of the condition after a negative test |
|---|---|---|---|
| 0.500% | 8.3% | 5.4% | 0.053% |
| 0.750% | 12.0% | 5.6% | 0.079% |
| 1.0%worked example | 15.4% | 5.9% | 0.106% |
| 1.3% | 18.6% | 6.1% | 0.133% |
| 1.5% | 21.5% | 6.3% | 0.160% |
Every input, tested
Which input moves the chance of the condition after a positive test most
Of the 3 inputs, prevalence (prior chance) moves the chance of the condition after a positive test most (6.6% across the range tested) and sensitivity moves it least (0.579%).
| Input | Tested from | To | Chance of the condition after a positive test at each end | Swing |
|---|---|---|---|---|
| Prevalence (prior chance) | 0.750% | 1.3% | 12.0% to 18.6% | 6.6% (43%) |
| False positive rate | 4.0% | 6.0% | 18.5% to 13.2% | 5.4% (35%) |
| Sensitivity | 88.0% | 92.0% | 15.1% to 15.7% | 0.579% (3.8%) |
Two variables at once
Chance of the condition after a positive test by prevalence (prior chance) and sensitivity
Across the grid the chance of the condition after a positive test runs from 8.0% to 22.3%. Moving prevalence (prior chance) from 0.500% to 1.5% shifts it by 13.2% at the middle column, and moving sensitivity from 86.0% to 94.0% shifts it by 1.2% at the middle row, so prevalence (prior chance) is the bigger lever here.
| Prevalence (prior chance) \ Sensitivity | 86.0% | 90.0% | 94.0% |
|---|---|---|---|
| 0.500% | 8.0% | 8.3% | 8.6% |
| 0.750% | 11.5% | 12.0% | 12.4% |
| 1.0% | 14.8% | 15.4% | 16.0% |
| 1.3% | 17.9% | 18.6% | 19.2% |
| 1.5% | 20.8% | 21.5% | 22.3% |
The highlighted cell is the worked example: 15.4%.
Step by step
The worked example, input by input
| Input | Value used | What it means |
|---|---|---|
| Prevalence (prior chance) | 1.0% | How common the condition is before any test. |
| Sensitivity | 90.0% | The chance the test is positive when the condition is present. |
| False positive rate | 5.0% | The chance the test is positive when the condition is absent. |
| Chance of the condition after a positive test | 15.4% | |
| Chance of a positive test overall | 5.9% | |
| Chance of the condition after a negative test | 0.106% | |
Inputs, definitions and assumptions
Prevalence (prior chance)
How common the condition is before any test. The prefilled worked-example value is 1.0%.
Sensitivity
The chance the test is positive when the condition is present. The prefilled worked-example value is 90.0%.
False positive rate
The chance the test is positive when the condition is absent. The prefilled worked-example value is 5.0%.
How to use this calculator
- 1Verify the inputs. Gather prevalence (prior chance), sensitivity and false positive rate from your own documents; the prefilled values are examples.
- 2Save a baseline. The worked example puts the chance of the condition after a positive test at 15.4%. Store your own version of it as Scenario A.
- 3Test one change. Start with prevalence (prior chance), the input with the biggest effect here: moving prevalence (prior chance) from 0.750% to 1.3% takes the chance of the condition after a positive test from 12.0% to 18.6%, a swing of 43% of the worked-example figure.
- 4Check the extremes. At half the example prevalence (prior chance) (0.500%) the chance of the condition after a positive test is 8.3%; at double (2.0%) it is 26.9%.
People also ask
Frequently asked questions
How do you calculate bayes' theorem?
P(condition | positive) = sensitivity × prevalence ÷ [sensitivity × prevalence + false positive rate × (1 − prevalence)]. Enter prevalence (prior chance) in percent, sensitivity in percent and false positive rate in percent (1 means 1%). At the worked-example inputs the chance of the condition after a positive test is 15.4%.
What does the bayes' theorem result mean?
See why a positive result on a good test can still mean a low chance of the condition when it is rare. At the worked-example inputs the chance of the condition after a positive test is 15.4%. It rises with prevalence (prior chance) and sensitivity and falls as false positive rate increases.
How much does prevalence (prior chance) change the chance of the condition after a positive test?
Holding every other input at the worked-example value, moving prevalence (prior chance) from 0.500% to 1.5% moves the chance of the condition after a positive test from 8.3% to 21.5%, a spread of 13.2%.
What are the limits of this bayes' theorem calculator?
These are standard formulas from introductory statistics. Real-world inference depends on assumptions, normality, independence, a genuinely random sample, that this calculator does not check for you. The tables on this page test prevalence (prior chance) only from 0.500% to 1.5%; a value outside that range is not tabulated here.
Which input moves the chance of the condition after a positive test most in the bayes' theorem calculator?
Ranked by how far each moves the chance of the condition after a positive test across the range tested: prevalence (prior chance) (6.6%, 43%), false positive rate (5.4%, 35%) and sensitivity (0.579%, 3.8%).
If I double prevalence (prior chance) in the bayes' theorem calculator, does the chance of the condition after a positive test double?
Doubling it from 1.0% to 2.0% takes the chance of the condition after a positive test from 15.4% to 26.9%, which is 1.75 times the worked-example figure. So it grows, but by less than double. Halving it to 0.500% gives 8.3%.
How much does sensitivity matter in the bayes' theorem calculator?
The worked example uses 90.0%. With the other inputs left at the worked example, moving sensitivity from 88.0% to 92.0% takes the chance of the condition after a positive test from 15.1% to 15.7%, a swing of 3.8% of the worked-example figure.
How much does false positive rate matter in the bayes' theorem calculator?
The worked example uses 5.0%. Holding every other input at its worked-example value, moving false positive rate from 4.0% to 6.0% takes the chance of the condition after a positive test from 18.5% to 13.2%, a swing of 35% of the worked-example figure.
Which inputs change the chance of a positive test overall in the bayes' theorem calculator?
At the worked-example inputs it is 5.9%. Prevalence (prior chance) takes it from 5.6% to 6.1%, sensitivity takes it from 5.8% to 5.9% and false positive rate takes it from 4.9% to 6.8%.
Which inputs change the chance of the condition after a negative test in the bayes' theorem calculator?
At the worked-example inputs it is 0.106%. Prevalence (prior chance) takes it from 0.079% to 0.133%, sensitivity takes it from 0.127% to 0.085% and false positive rate takes it from 0.105% to 0.107%.
What does a 95% confidence interval actually mean?
It means that if you repeated the sampling process many times and built an interval the same way each time, about 95% of those intervals would contain the true population value. It does NOT mean there's a 95% probability the true value falls in this one specific interval: a common and understandable misreading.
Why do smaller margins of error need disproportionately larger samples?
Required sample size grows with the square of how small you want the margin of error to be. Cutting the margin of error in half roughly quadruples the sample size needed, which is why polls rarely promise a margin under 2-3% even with very large samples.
Sources and evidence
Free Calculators Online is independent and is not affiliated with or endorsed by the source organizations. Educational estimates only.