Bayes' Theorem and Positive Tests

A note before we start: this piece is about how to read test results, not medical advice. Real tests vary by type, age and setting, and a doctor weighs far more than one number. If you get a positive result, ask your doctor what it means for you.

In 2007 the psychologist Gerd Gigerenzer put a version of this problem to about 160 gynaecologists. Imagine screening women for breast cancer. About 1% of them have it. If a woman has it, the mammogram is positive 90% of the time. If she doesn't, it is still positive 9% of the time. A woman tests positive. How likely is it that she has cancer?

The most common answer was 90%. The right answer is about 9%, roughly 1 in 10. These were specialists who give such results to patients, and the error is not about medicine. It is about a piece of arithmetic almost nobody is taught to see.

Turn percentages into people

Gigerenzer's fix is simple: stop thinking in percentages and think in people. Picture 1,000 women.

	Test positive	Test negative
10 have cancer (1%)	9	1
990 don't	about 89	about 901

Now count the positives: 9 real cases plus about 89 false alarms, about 98 in all. Only 9 of those 98 have cancer. That's 9.2%. The test isn't bad. The problem is that healthy women vastly outnumber sick ones, so even a small false-alarm rate, applied to a big crowd, swamps the real cases.

A positive result doesn't tell you that you have the disease. It tells you that you've moved into a smaller, more worried crowd. What matters is who else is in that crowd.

Try it: the positive-test calculator

Play with it. Drag prevalence up and the same test becomes trustworthy. Drag the false-positive rate down and watch the orange dots vanish. The "Same test, high-risk patient" button is the most important lesson here. Someone who already has symptoms belongs to a very different starting crowd from someone at a routine screening. If her starting chance were, say, 30% (an illustrative figure), the very same positive result would mean about 81% rather than 9%.

The rule itself

What you just did by counting has a name. Bayes' theorem says, in words: the chance you have the disease, given a positive test, equals the share of all positive tests that come from sick people. In symbols:

P(sick | positive) = P(positive | sick) × P(sick) ÷ P(positive)

The left side is what you want to know. The right side contains what the test-maker tells you (sensitivity) and the thing everybody forgets: P(sick), the base rate, how common the condition was before you took the test. Leave it out and you get the gynaecologists' 90%.

The version almost nobody learns: odds

There is a cleaner way to write the rule, and it is the version that changes how you think. Use odds instead of probabilities.

Odds after the evidence = odds before × the strength of the evidence.

The strength of a positive test is its likelihood ratio: how much more often it happens to sick people than to healthy ones. Here that is 90% ÷ 9% = 10. Before the test, the odds were 1 to 99 (10 sick women against 990 healthy ones). Multiply by 10 and you get 10 to 99, which is 9.2%. Same answer, no table needed.

This form tells you something deep. Evidence doesn't replace what you knew before. It multiplies it. A test that multiplies the odds by 10 is strong evidence, but ten times "very unlikely" is still "unlikely". The calculator's readout shows both steps.

Two positives are much stronger than one

Suppose the woman takes a second test whose errors are independent of the first. Now her starting point is no longer 1%. It is 9.2%, the result of the first test. Multiply those odds by 10 again and she is at about 50%. A third positive would take her to about 91%. Each piece of evidence starts from wherever the last one left you. That is what learning looks like when it's done properly.

The catch is the word independent. Repeating the same test on the same person often repeats the same error (the same harmless shadow on a scan, say). That is why a follow-up test that works in a different way tells you more than simply repeating the first one.

A rule found twice

Thomas Bayes died in Tunbridge Wells, in England, in 1761 without publishing it. His friend Richard Price found the essay among his papers, edited it, added to it and sent it to the Royal Society, which published it in 1763. How much of the final text is Price's is still debated. In 1774 the French mathematician Pierre-Simon Laplace arrived at the same idea, apparently independently, and developed it into a general method. Historians often say Bayes deserves the name for being first and Laplace deserves the credit for making it a tool.

Why this is more than a medical trick

Bayes' rule is the arithmetic of changing your mind. Every belief starts somewhere, every piece of evidence pushes it by a factor, and the result is your new starting point. It quietly explains a lot. It explains why a surprising claim needs strong evidence: when the starting odds are tiny, a modest multiplier leaves them small. It explains why two honest people with different starting points can see the same evidence and still disagree, and why they should end up closer together than they began.

It also contains a small warning. If your starting probability is exactly zero, no evidence can ever move it, because anything times zero is zero. The one belief Bayes' rule cannot repair is total certainty.