Posterior odds = 0.2, i.e. 1:5 (about 16.7% as a probability).
- Prior odds: 0.01 / 0.99 ≈ 0.0101 (1:99).
- Likelihood ratio: 0.99 / 0.05 = 19.8.
- Posterior odds: 0.0101 × 19.8 = 0.2.
Converting back to a probability as a sanity check: P = odds / (1 + odds) = 0.2 / 1.2 ≈ 0.167 — matching what you’d get grinding through lesson 6’s full formula with the same numbers.
Why the denominator disappears. Bayes’ theorem in probability form is:
P(H | E) = P(E | H)·P(H) / P(E)
The awkward part is always P(E) — “the probability of the evidence, period,” which usually means expanding it across every hypothesis (lesson 6’s full denominator). But take the ratio of the theorem for H and for its negation ¬H, and P(E) is the same term top and bottom — it cancels:
P(H|E) / P(¬H|E) = [P(E|H) / P(E|¬H)] × [P(H) / P(¬H)]
Read left to right, that’s exactly posterior odds = likelihood ratio × prior odds. You never had to compute P(E) at all — the thing that made lessons 6-8 require a full population count.
What the likelihood ratio is, intuitively: how many times more consistent the evidence is with H than with ¬H. LR = 1 means the evidence is neutral — it doesn’t move you. LR > 1 pushes toward H; LR < 1 pushes toward ¬H (a negative test result would have an LR below 1, shrinking the odds instead of growing them — try that as an exercise: LR = 0.01/0.95 for a negative result in this same scenario). LR = 19.8 here means “this test result is about 20 times more likely under disease than not” — that’s the entire strength of the evidence, in one number, independent of the prior it gets multiplied against.
Why this form earns its keep: it separates what you believed before (prior odds) from how strong this evidence is (LR) cleanly enough that you can cache the LR and reuse it — swap in a different prior (a different patient, a different population) without recomputing anything about the test itself. That reusability is exactly what tomorrow’s lesson needs: chaining a second piece of evidence is just multiplying in a second likelihood ratio.
Where this goes: two tests instead of one — when can you legally multiply their likelihood ratios together, and when does that silently assume something false?