Thought Toys · Chance & inference · Exhibit 09
A disease affects one person in a hundred. There's a test that's 90% accurate, and yours comes back positive. It feels like a 90% verdict — brace yourself. But the honest answer is closer to 9%. The test didn't lie; it's just that when a disease is rare, the few false alarms badly outnumber the real cases. Count it out, one thousand people at a time, and the paradox dissolves.
A positive test means a 9% chance you're sick
The thousand dots are a whole town. Up top, in gold, sit the people who are genuinely sick. Most of them — nine in ten with this test — get correctly flagged (bright gold); a few slip through missed (faint gold). Then come the red dots: people who are perfectly healthy but got a positive result anyway. There aren't many per person — the test is right about 9 healthy people in 10 — but there are so many healthy people that even that small slip-up rate produces a big red crowd.
Now the only question that matters after your positive result: among everyone the test flagged — the gold-real plus the red false alarms — what share are actually sick? That's the bar below the grid. When the disease is rare, the red badly outnumbers the gold, so a positive test is mostly a false alarm and your real odds stay low. The test was never wrong about its accuracy; your gut just quietly ignored how rare the thing was to begin with. That blind spot has a name: base-rate neglect — trusting a test's accuracy while forgetting how rare the thing it's testing for actually is.
Slide how common the disease is upward and watch the red shrink against the gold — for a common condition, the same test suddenly becomes trustworthy. That's the whole lesson of Bayes: a piece of evidence doesn't hand you the answer, it updates what you already knew. Start rare, end rare-ish; start likely, end likely. The prior always gets a vote.
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