Lesson 6 derived Bayes’ theorem from the joint. There’s a second form of it — the odds form — that turns the whole update into one multiplication, no denominator required.
First, odds are just a different way to say a probability: if P(disease) = 0.01, the odds of disease are P(disease) / P(no disease) = 0.01 / 0.99 ≈ 1:99. Odds and probability carry the same information; odds are just “for against how many for” instead of “fraction out of the whole”.
The likelihood ratio (LR) for a piece of evidence E is:
LR = P(E | disease) / P(E | no disease)
— how much more likely the evidence is if the hypothesis is true, versus if it’s false. And here is the odds form of Bayes’ theorem:
posterior odds = prior odds × likelihood ratio
No P(E) in sight. Lesson 7’s rare-disease test, rebuilt as this one multiplication:
- Prior odds: the disease has a 1% base rate, so prior odds = 0.01 / 0.99 ≈ 0.0101.
- The test: 99% sensitivity (P(positive | disease) = 0.99) and a 5% false-positive rate (P(positive | no disease) = 0.05).
- Likelihood ratio for a positive result: LR = 0.99 / 0.05 = 19.8.
Multiply prior odds by the likelihood ratio to get the posterior odds of disease given a positive test. (Odds, not probability — you can convert back with P = odds / (1 + odds) if you want to check it against lesson 7’s answer, but the question only asks for the odds.)