False Positive Calculator

Find the false positive rate and the probability that a positive test result is actually wrong, from prevalence, sensitivity, and specificity using Bayes' theorem.

Quick Facts

Method
FPR = 1 − specificity; P(false alarm | positive) via Bayes' theorem
Combines base rate with test accuracy to expose the base-rate effect.

Your Results

Calculated
P(false positive | positive)
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Chance a positive result is a false alarm
False positive rate
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1 − specificity
Positive predictive value
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Chance a positive is truly affected
Expected false positives
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Per test population

Ready

Enter prevalence, sensitivity, and specificity, then calculate.

How the False Positive Calculator works

A "false positive" is a negative case that a test wrongly flags as positive — a healthy person told they may be sick, a legitimate email marked as spam, a clean sample flagged by a screen. This calculator answers two related questions: how often the test raises a false alarm (the false positive rate), and — more usefully — given that you got a positive result, how likely it is to actually be wrong. The second question depends heavily on how common the condition is, and the answer is often far more alarming than people expect.

The two core formulas

The false positive rate (FPR) is a fixed property of the test itself:

FPR = FP / (FP + TN) = 1 − specificity

A test with 95% specificity has an FPR of 5%: five of every hundred truly-negative people test positive anyway. But the FPR alone does not tell you whether your positive result is real. For that you need Bayes' theorem, which folds in the prevalence (base rate) and the sensitivity:

P(false positive | positive) = [(1 − prevalence) × FPR] / [(1 − prevalence) × FPR + prevalence × sensitivity]

The complement of that quantity is the positive predictive value (PPV) — the chance a positive result is a true positive: PPV = 1 − P(false positive | positive).

Why the base rate dominates

When a condition is rare, the pool of healthy people is enormous compared with the pool of affected people. Even a small false positive rate applied to that huge healthy pool can generate more false alarms than the test generates true positives from the small affected pool. Take a 1% prevalence, 90% sensitivity, and 90% specificity: out of 10,000 people, 100 are affected and 90 test positive correctly; 9,900 are healthy and 990 (10%) test positive falsely. Of the 1,080 total positives, 990 are false — about 91.7%. The test is "90% accurate" yet a positive result is wrong more than nine times in ten. That is the base-rate effect, and it is the single most important idea this calculator illustrates.

Common reference points

  • At 50% prevalence with a 5% FPR and 95% sensitivity, a positive is right about 95% of the time — the base rate barely hurts you.
  • At 0.1% prevalence (1 in 1,000) with a 99% specificity and 99% sensitivity, a positive is still wrong roughly 91% of the time.
  • Raising specificity is usually the fastest way to cut false positives, because P(false positive | positive) scales with (1 − specificity).
  • Confirmatory two-stage testing (a cheap sensitive screen followed by a specific confirmatory test) is the standard fix for low-prevalence screening.

Frequently Asked Questions

What is the difference between the false positive rate and the false discovery rate?
The false positive rate (FPR = 1 − specificity) is the chance a negative case tests positive. The false discovery rate is the chance a positive result is actually negative — that's the "P(false positive | positive)" this calculator reports as its headline number. FPR is a property of the test; the false discovery rate also depends on prevalence.
Why is my positive result probably wrong even with an accurate test?
Because the condition is rare. When only 1% of people have a condition, the 99% who don't vastly outnumber them, so even a few percent of false positives among that huge group swamp the true positives. This base-rate effect means a highly accurate test can still yield mostly-false positives when it screens a low-prevalence population.
How do I lower the false positive rate?
Increase specificity — the false positive rate is exactly 1 − specificity. In practice that means a stricter positivity threshold, a better assay, or a confirmatory second test on anyone who initially screens positive. Raising the threshold reduces false positives but usually costs some sensitivity, so the two are traded off along a ROC curve.

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