Right ninety-five times in a hundred, and most of its yeses are wrong
A test finds 95 in every 100 people who have a disease and clears 95 in every 100 who do not. It is used on a town where one person in a hundred has it. Somebody tests positive. The chance that they are sick is not 95%. It is — — about one in six — because of every 10,000 people tested the test says yes to —, and only — of those are sick. The rest are the five in a hundred it gets wrong, taken from a crowd ninety-nine times the size.
Ten thousand people, one square each
Every square is a person. They are sorted rather than scattered, so that each kind of person is one block you can see the size of: first the sick people the test finds, then the well people it says yes to anyway, then the sick people it misses, then everybody it rightly clears. The first two blocks are everybody with a positive result, and the question a positive result asks is how much of that is the first block — so those two are drawn again above the town, as large as they will fit.
The squares are whole people, so the four blocks are rounded to add up to ten thousand, the largest remainders rounding up. The readouts are not rounded: an expected count of 0.95 people is printed as 0.95.
Which dial matters
When a disease is rare, almost everybody tested is well, so the well people the test gets wrong are a small fraction of a very large number. That makes the second dial — how many well people it clears — the one that decides what a yes is worth, and the first dial nearly irrelevant.
The same test, anywhere
A test has no fixed chance of being right when it says yes. Carry this one from town to town and the chance runs from almost nothing to almost certain, according to nothing but how common the disease is where it is used. The blue curve is one positive result; the amber curve is two, from two tests that are independent of each other.
Test again
A positive result does not settle anything, but it changes who you are: you are no longer somebody from a town where one in a hundred is sick, you are somebody from the block of people who tested positive, and in that block — Test that block again and the same arithmetic starts from there. Each positive multiplies the odds of being sick by the same number — the sick people it finds over the well people it wrongly says yes to, —
Independent is doing a great deal of work in that table. A second run of the same test on the same sample repeats whatever fooled it the first time, and then the second positive is worth much less than the table says. The arithmetic is about a second test that errs for different reasons.
Three routes to one number
The page works the same answer out three ways and checks below that they agree at every stop of every dial. Counting people, as above: the sick who test positive over everybody who tests positive. Bayes’ rule, written as chances: P(sick | yes) = P(yes | sick) P(sick) ⁄ P(yes). And odds, which is the form worth remembering because it can be done in your head: —