PS
LabManifestServicesProjectsCollaborationAssociationContact
Grasping
Probability · Diagnosis

A positive test isn't a diagnosis.

Most doctors and most patients get this wrong. The same test number — say a 90% sensitive mammogram — gives wildly different answers depending on how common the disease is. Drag the sliders below: this is Bayes' theorem at work.

P(disease · positive test)
7.1%

Of all people who test positive, this is the share that actually has the disease. The other share is false alarm.

Per 10,000 people tested
True positive
68
0.68%
Missed (false negative)
12
0.12%
False alarm (false positive)
893
8.9%
True negative
9’027
90.3%
↑ Has disease
↑ Does not have disease
Of 961 positive tests
68
893
True positiveFalse alarm (false positive)
Real-world setups
The formula

Bayes' theorem, written out.

Posterior equals likelihood times prior, divided by the total probability of the evidence. The unintuitive part: the denominator is mostly false positives when the disease is rare.

P(D | +) = P(+ | D) · P(D) / P(+)
= P(+ | D) · P(D) / [ P(+ | D) · P(D) + P(+ | ¬D) · P(¬D) ]
P(D) · Prevalence
P(+|D) · Sensitivity
P(−|¬D) · Specificity
P(D|+) · Posterior
Story

Why the math is so counter-intuitive.

  1. 1763 — a posthumous essay.

    Thomas Bayes, a Presbyterian minister and amateur mathematician, leaves behind an unpublished essay. Richard Price publishes it two years after Bayes' death. It introduces the idea of updating beliefs with evidence, mathematically. For nearly two centuries it sits at the edge of statistics.

  2. 1978 — doctors fail the test.

    David Eddy presents the classic mammography problem to 100 American physicians: 1% prevalence, 80% sensitivity, 90% specificity. Most say the chance of cancer given a positive is around 75%. Correct answer: under 8%. The error isn't carelessness — it's that human intuition ignores the base rate.

  3. 1995 — natural frequencies fix it.

    Gigerenzer and Hoffrage rephrase the problem in natural frequencies ("of 1000 women, 10 have cancer; 8 of them get a positive test; of the remaining 990, 99 also get a positive test — what's the chance?"). Suddenly doctors get it right. The math is identical, only the framing changed.

  4. Today — most screening is rarity-dominated.

    For rare conditions even a near-perfect test produces mostly false alarms. This is why population screening is debated: not because the test is bad, but because base rates are low. Bayes' theorem doesn't argue against testing — it argues for understanding what a positive result means before reacting.

Sources