Each of the 6,400 cubes below is one simulated person. A cube is lifted and coloured when that person's test comes back positive; the raised layer is the entire "positive-test pool" the article talks about — and most of it is false alarms when the disease is rare.
Bayes' theorem: P(A|B) = P(B|A)·P(A) / P(B)
Applied here, with A = Disease, B = Positive test:
P(Disease|Positive) = P(Positive|Disease)·P(Disease)
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P(Positive|Disease)P(Disease) + P(Positive|No disease)P(No disease)
Natural-frequency version (out of N people):
true positives = N × prevalence × sensitivity
false positives = N × (1 − prevalence) × (1 − specificity)
posterior = true positives / (true positives + false positives)
- Prevalence — how rare the disease is; lowering it shrinks the green "true positive" layer while the orange "false positive" layer (drawn from the much larger healthy majority) barely changes.
- Sensitivity — the fraction of sick people the test catches; raising it grows the green layer and shrinks the red "false negative" layer.
- Specificity — the fraction of healthy people correctly cleared; even a small drop here floods the raised layer with orange false positives once the population is large.
- Retest button — feeds this round's posterior back in as next round's prior, the same sequential-updating idea used when a patient is tested twice.
With the classic textbook numbers — 1% prevalence, 90% sensitivity, 95% specificity — a positive result only raises the probability of disease to about 15.4%, because the 495 false alarms from the healthy 99% swamp the 90 true positives from the sick 1%.