08 / 10 · Probability
Bayes' theorem
What a positive test really means
A disease affects 1 in 100 people. A test catches 90% of sick people, but also wrongly flags 9% of healthy people. You test positive. How likely are you to be sick?
Most people say 90%. The real answer is about 9%. The easiest way to see why is to imagine 1,000 people instead of thinking in percentages.
P(sick | positive) = true positives ÷ all positives
= 9 ÷ (9 + 89) ≈ 0.09Try it · Each square is one of 1,000 people. Move the sliders and see how the orange positives split between sick and healthy.
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Of 1,000 people, 10 are sick and the test catches 9 of them. The other 990 are healthy, but 9% of them, about 89, also test positive. So 98 people test positive, and only 9 of them are sick: 9 ÷ 98 ≈ 9%.
The test did its job: your chance went from 1% to 9%. But because the disease is rare, false alarms from the huge healthy group outnumber true cases. Bayes' theorem is the rule for updating a probability when new evidence arrives.
Key ideas
Words worth remembering
- Base rate
- How common something is before you see any evidence.
- Conditional probability
- P(A | B): the probability of A once you know B happened.
- False positive
- A test that says yes when the true answer is no.
Think about it
What would you need to change to make a positive result 90% reliable?