Bayes: how likely is illness after a positive test?
The classic of conditional probability: for rare diseases a positive test is often less telling than it seems.
Worked example
A disease affects 5% of people. A test detects the ill with 95% (sensitivity) and the healthy with 95% (specificity). What is P(ill | positive test)?
Worked solution
- 1.P(+) = 0.95 · 0.05 + 0.05 · 0.95 = 0.0950
- 2.Bayes: P(ill | +) = P(+ | ill) · P(ill) / P(+) = 0.0475 / 0.0950 = 0.5000
Typical mistake: Confusing P(ill | +) with the sensitivity P(+ | ill).
Your turn
Three problems of exactly this type, each time with new numbers. Comma or point; fractions like 1/6 work too.
Task
A disease affects 5% of people. A test detects the ill with 95% (sensitivity) and the healthy with 95% (specificity). What is P(ill | positive test)?
P (4 decimal places)
Comma or point; fractions like 1/6 work too.
Formulas for this topic
- P(A | B) = P(A ∩ B) / P(B)
- Total probability: P(B) = P(B | A) · P(A) + P(B | Ā) · P(Ā)
- Bayes: P(A | B) = P(B | A) · P(A) / P(B)
Typical mistake: Confusing P(ill | +) with the sensitivity P(+ | ill).
Frequently asked questions
What is the typical mistake in "Bayes: how likely is illness after a positive test?"?
Confusing P(ill | +) with the sensitivity P(+ | ill).
Why is P(ill | positive) often small?
For rare diseases there are far more healthy people, so even a few false positives outnumber the true positives.