Posterior odds ≈ 0.3, i.e. about 3:10 (roughly 23% as a probability).
- Prior odds: 1/999 ≈ 0.001.
- LR_A × LR_B = 20 × 15 = 300.
- Posterior odds: 0.001 × 300 ≈ 0.3.
Two positive tests, each individually unimpressive-sounding (LR 20, LR 15 — nowhere near “certain”), combine to push a 1-in-1,000 prior up to roughly 1-in-4. That’s the payoff of odds form from last lesson: sequential evidence is just sequential multiplication, when you’re allowed to multiply.
The assumption doing all the work: conditional independence given disease status. Formally:
P(A positive, B positive | disease) = P(A positive | disease) × P(B positive | disease)
and the same equation with ¬disease in place of disease. In words: once you already know whether someone has the disease, learning Test A’s result gives you zero extra information about what Test B will say. Their errors and their signals both have separate causes.
What if that assumption is false? Say the two tests instead shared a hidden cause of false positives — for instance, both assays cross-react with the same common, harmless antibody that some healthy people happen to carry. Then among the disease-free, a false positive on Test A makes a false positive on Test B more likely too (same underlying cause), not independent of it. Naively multiplying LR_A × LR_B then overstates the combined evidence — you’re partly counting the same fact twice, dressed up as two facts. (This is the same mistake lesson 4 introduced in general — “assuming independence when evidence is actually correlated” — now caught in the specific act of compounding.) The fix isn’t algebraic, it’s empirical: measure the joint likelihood ratio directly, P(A+, B+ | disease) / P(A+, B+ | ¬disease), rather than assuming it factors.
How to tell, in practice: ask whether the tests could go wrong for the same reason. Two tests built on the same instrument, the same sample, the same underlying biomarker, or scored by the same rater are the tell-tale cases (in forecasting: two “independent” pundits who both read the same wire report aren’t independent evidence, they’re one piece of evidence twice). Two tests built on genuinely different mechanisms — a chemical assay and a genetic panel, say — are the clean case, and clean cases are what medicine and forensics reach for when they can.
Where this goes: so far every lesson has updated a belief about a fixed, binary hypothesis (disease or not, guilty or not). Stage 2 opens the belief up into a full distribution — not “is the rate 10%?” but “here’s the whole plausibility curve over every rate from 0 to 100%,” updated by data the same multiplicative way.