A positive test
A disease affects of a population. A test detects it in of ill people and correctly returns negative for of healthy people.
A person picked at random tests positive. What is the probability that they have the disease?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Imagine people. How many are ill and test positive? How many are healthy and test positive?
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. A positive result can come from the few ill people or from the many healthy people who get a false positive. Here the two groups turn out to be the same size.
Step 1: count in a population of .
- Ill: people, of whom test positive.
- Healthy: people, of whom , that is , test positive.
Step 2. There are positive results, of them from ill people:
With Bayes' rule.
The lesson. A accurate test still gives a coin flip here, because the disease is rare: the false positives from the large healthy group match the true positives. Testing again (independently) would move the odds from to .