The 99 Percent Test That Is Usually Wrong

A thousand people walk into the picture, every one of them gets tested, and the board pulls everyone who tested positive into a single pile so you can count who is actually standing in it.

Before you start. You need to be able to work out a percent of a round number, the kind of arithmetic you’d use to check a bill. That is the whole of the maths in the story half of this lesson.

The idea in 30 seconds

A test answers “how often am I right about someone who has it,” not “how likely is this person to have it” - different questions wearing the same-looking number.
The rarer the condition, the more a tiny error rate matters, because the healthy crowd it can misfire on is so much bigger than the sick one.
Out of 1,000 people tested for a 1-in-1,000 condition with a 99%/1% test, about 11 come back positive and only 1 of them really has it: about 9 percent.
That share, positives that are actually real, is called positive predictive value, and it depends on the room the test is pointed at, not just the test.
Bayes’ theorem is the general arithmetic behind that counting: start from how common the thing is, then update with what the test told you.
The same test can be nearly useless as a screen for everybody and excellent once pointed at people already likely to have the thing.

Read the story · Skip to the numbers