≈ 0.047 — under 5%. A positive result on a “99% accurate” test for this condition means you’re still probably fine.
The count, with 100,000 people:
Have it: 100 people (1 in 1,000). Of them, 99 test positive. → 99 true positives
Don’t have it: 99,900 people. Of them, 2% test positive. → 1,998 false positives
Everyone who tested positive: 99 + 1,998 = 2,097. Actually sick: 99.
P(condition | positive) = 99 / 2,097 ≈ 0.047
Same shape as the spam problem — prior × likelihood on each branch, then “what fraction of the evidence-pool came from the hypothesis branch?” What changed is only the prior: 0.4 became 0.001, and the posterior collapsed from 86% to 5%.
Why intuition fails: “the test is 99% accurate” describes P(positive | condition) — how the test behaves given the truth. The question asks P(condition | positive) — the reverse conditional, which depends on something the accuracy number doesn’t contain: how rare the condition is. The healthy group is 999× larger than the sick group, so even its measly 2% error rate produces 20× more false positives than the sick group produces true ones. Ignoring the prior’s size when reversing a conditional is called base-rate neglect, and it’s the most common quantitative reasoning error in professional life. Your monitoring system knows it well: when real incidents are rare, an alert that’s “99% accurate” pages you mostly for ghosts — the alerting version of this arithmetic is why alert fatigue is a base-rate problem, not a diligence problem.
What would rescue the test? Two things, and they’re instructive:
- A higher prior. Test people with symptoms (prior 1-in-10 instead of 1-in-1,000) and the same test gives P(condition | positive) ≈ 85%. Screening and diagnosis are different régimes.
- A second, independent test. A positive moved you from 0.1% to 4.7% — a ~50× boost in odds. Another independent positive would boost again, to ≈ 70%. Independent is doing heavy lifting there — lesson 4’s warning applies (same lab, same failure mode → no second boost).
That “boost” language — each piece of evidence multiplying your odds by a fixed factor — is not a metaphor. It’s a theorem, it makes sequential updating nearly mental arithmetic, and it’s two lessons away.