What does a positive really mean?

A test that's 99% accurate sounds airtight — until you test for something rare. Then most positives are false alarms, because a tiny error rate applied to a huge healthy majority outnumbers the few real cases. Feed in how common the condition is and how good the test is, and read off what a positive actually means.

The test
% of peoplehave itPrevalence, the base rate: how common the condition is in the group being tested. The single biggest lever on what a positive means — and the one intuition ignores.. A testcatchesSensitivity: of the people who do have it, the share the test correctly flags. The true-positive rate.% of them andclearsSpecificity: of the people who don't have it, the share the test correctly clears. The true-negative rate; 100% minus it is the false-alarm rate.% of the rest.
50%chance it's realPositive predictive value: P(has it | tested positive). The chance a positive result is real — the thing people confuse with the test's accuracy.

Test positive and there's a 50% chance you actually have it — about 1 false alarm for every real case.

50%positive is real
100%negative is clearNegative predictive value: P(clear | tested negative). The chance a clean result really is clean — usually the reassuring one when the condition is rare.
50%positives are false
Counter-intuitively, a positive is more likely wrong than right. Of about 198 positives, 99 are false alarms and only 99 are real — because the condition is rare, the few real cases are swamped by a small slip-up rate applied to the huge healthy majority.
Out of
people
Test +
Test −
100 have it
99true positive
1missed
9,900 don't
99false alarm
9,801cleared
Of 198 positives:
99 real
99 false
As the condition gets rarer
PrevalenceA positive is real
0.1%
9%
0.5%
33.2%
1%now
50%
2%
66.9%
5%
83.9%
10%
91.7%
25%
97.1%
50%
99%

Same test, every row — only the base rate changes. The rarer the condition, the further a positive tips toward a false alarm, because the test's small slip-up rate is applied to an ever-larger healthy majority. That's the base-rate fallacy: accuracy isn't the answer to "does a positive mean I have it?".

based on the base-rate fallacy