Bayes' theorem: when a positive test is not enough

Why a good test can still give mostly false alarms, shown with a table of 100,000 people. About 9 minutes, A-level.

What you will do

1. Test results are not certainties. A medical test can be wrong two ways: it can miss someone who has the condition (sensitivity measures how rarely) or raise a false alarm on a healthy person (specificity measures how rarely).
2. A crowd of 100,000. Probabilities are easier as counts. Imagine 100,000 people and split them using each percentage.
3. The surprising answer. Most positives here are false positives, because there are so many more healthy people than sick ones.
4. When the condition is common. Raise the prior and the picture changes completely: now a positive is much more likely to be real.
5. Bayes' theorem. The table and the formula are the same thing: P(D +) = P(+ D)P(D)P(+ D)P(D) + P(+ D)P( D).