The False Positive Paradox

The problem
Lab resultPOSITIVEP(sick | +) = ?base rate 1/100 · sens 99% · spec 90%

A disease affects 1 in 100 people. A test for it has:

  • 99% sensitivity — if you have the disease, it catches it 99% of the time.
  • 90% specificity — if you don't have it, the test correctly says so 90% of the time (10% false positive rate).

You take the test. It comes back positive. What's the probability you actually have the disease?

Tempting (but wrong)
"The test is 99% accurate, so I'm 99% sick"99%sensitivityspecificity 90%ignores base rate

The instinct is "the test is 99% accurate, so I'm 99% likely to be sick."

Studies have asked doctors this exact question — including doctors actively interpreting screening results — and the modal answer is somewhere between 70% and 95%. The right answer isn't close.