BAYES — how a rare truth hides inside a reliable test: the false-positive paradox as a live area diagram

◧ BAYES
The false-positive paradox

Screen a hundred thousand people for something rare. The test barely errs — yet the pile of positive results fills, overwhelmingly, with the healthy. Drag the base rate below and watch belief re-weigh itself: it is rarity, not inaccuracy, doing the damage.

One hundred thousand people, screened once

Population
Proportional-area partition of a screened population The thin left column is the sick minority; the wide right column is the healthy majority. The coloured top band of each column is who the test flags as positive. Sick 0.30% Healthy 99.70% Tested +
If your result is positive,
your chance of actually being sick
23%
More than three in four of these positive results — 77 in 100 — belong to people who are perfectly well.
Everyone the test flagged1,294 people
Composition of positive results: true positives versus false positives even odds
297 truly sick · 997 false alarms
Re-derived, every frame
P(sick | +) =
  Se·p
─────────────────
 Se·p + (1−Sp)(1−p)
= 0.00297 / 0.01294
= 22.9%
A positive result is not a diagnosis. It is a prior, dragged toward the truth by exactly how rare the truth is.

01 / Why the alarm lies

Sensitivity and specificity describe the test. They say nothing about you until the base rate speaks. When the condition is rare, the vast healthy majority — multiplied by even a one-percent error — produces more false positives than there are true cases to find.

02 / The prior does the work

Bayes' rule folds the base rate p into the verdict. Shift it and the posterior swings from near-certainty to near-nothing without the test changing at all. Evidence updates belief; it does not replace it.

03 / Read the area, not the accuracy

Trust the diagram over the percentage. The blue sliver of the flagged bar is every person the screen was right about; the red is the crowd it frightened for nothing. Rarity sets that ratio — drag the base rate up and watch the blue reclaim the bar.