The 99%-accurate test that’s a coin flip
Micro-lessons
A test is 99% accurate. You test positive. How worried should you be? For a rare disease the answer is often about 50/50 — and the reason is the number of healthy people, not the quality of the test.
Count people, not probabilities
Take 1000 people and a disease that afflicts 1% of them — 10 sick, 990 healthy. A test that catches 99% of the sick flags about 10 true positives. But at 99% specificity it also wrongly flags 1% of the 990 healthy — another ≈ 10 false positives. Twenty positive results, half of them healthy. A positive test is a coin flip.
Turn the knobs
Each square is one of 1000 people. Drag the base rate down and watch the red false-positives swamp the true positives — that ratio is your answer:
False positives are drawn from the large healthy pool, true positives from the small sick one. When the disease is rare, even a tiny false-positive rate times a big number rivals every true positive. The answer is always the same fraction — P(sick | +) = TP / (TP + FP) — and the prevalence is what tips it.
This is just Bayes
That fraction is Bayes’ theorem wearing work clothes. The prevalence is the prior; the test result is the evidence; the recomputed fraction is the posterior. It’s why screening a whole asymptomatic population needs punishing specificity, not just “accuracy,” and why a spam filter that’s 99% accurate still buries real mail if spam is rare in your inbox.
Say it, don’t just nod
A disease hits 0.1% of people; the test is 99% specific and catches every real case. Of 100 people who test positive, roughly how many are actually sick?
Per 100,000 people: 100 sick (all caught → 100 true positives) and 99,900 healthy, of whom 1% false-positive → ≈ 999. So P(sick | +) = 100 / (100 + 999) ≈ 9%.
Of 100 positives, only about 9 are truly sick — even with a perfect catch rate — because the 1% error on a huge healthy population dwarfs the handful of real cases.