You tested
positive.
Now what?

Here is a question that has been put to thousands of doctors, and most of them get it wrong. A woman with no symptoms has a routine mammogram. The test is good: it catches ninety per cent of real cancers, and it correctly clears ninety-one per cent of healthy women. In her age group, about one woman in a hundred has breast cancer. Her result comes back positive. What is the chance she actually has cancer?

Before you read another line, commit to an answer. Not a calculation — a gut feeling. Then we'll count it out.

01 Count the women, not the percentages

Percentages are slippery. The cleaner way to think about a test is to line up a thousand real people and follow what happens to each of them. So here are a thousand women from her age group. Watch where they land.

Of the thousand, about ten actually have cancer — that's the one-per-cent base rate. The test catches ninety per cent of them, so nine test positive and one slips through. The other 990 are healthy — but the test wrongly flags nine per cent of them, which is 89 women who get a frightening positive result despite being perfectly well. Drag the sliders and watch the four groups move.

02 Why the intuition fails

Look at what just happened. Nine genuinely ill women test positive. But eighty-nine healthy women also test positive, because nine per cent of a large number is itself a large number. The false alarms come from the huge healthy majority, and they swamp the handful of true positives. So a positive result is dominated not by the sick, but by the well who happened to trip the test.

That is the whole trap. We hear "ninety per cent accurate" and quietly assume a positive result is ninety per cent likely to be real. But the test's accuracy and the chance you're ill given a positive are two different numbers, and when the disease is rare they can be worlds apart. The rarer the condition, the more the false alarms dominate — which is exactly why a very good test can still be mostly wrong when it fires.

03 Bayes' theorem is just the bookkeeping

You didn't need a formula to get the right answer — you just counted dots. Bayes' theorem is nothing more than that counting, written down. The chance you're ill given a positive result is the ill-and-positive group divided by everyone who tests positive:

Every piece maps onto a block of dots. The top is the teal group; the bottom adds in the marigold false alarms. The base rate — that 0.01 — is doing the heavy lifting: shrink it and the teal block shrinks with it, while the false alarms barely budge. That's the base rate fallacy in one line: ignore the prior probability and the maths quietly falls apart.

04 Testing twice: your answer becomes the next question

If a single positive only gets us to nine per cent, what should the woman do? Not panic — but not ignore it either. She should test again. And here's the elegant part: today's answer becomes tomorrow's starting point. Her nine per cent is no longer a wild one-per-cent prior; it's a considered belief, and a second independent test updates it further. Statisticians call this Bayesian updating: the posterior from one test is the prior for the next.

Two positives in a row and she's near a coin-flip; three and it's serious. This is why screening programmes rarely act on a single positive — they recall you for a second look. Each test is not a verdict but an update, nudging a probability up or down. Notice too that the same machinery runs in reverse: a negative after a positive drags her belief back toward healthy, because most positives were false alarms to begin with. It's the same trick behind the Monty Hall problem: the host opening a door is new information, and the honest move is to let it rewrite your odds rather than cling to your first guess.

05 What screening actually costs

So far this looks like a maths puzzle. It isn't. Those eighty-nine false alarms are real women who spend a week believing they may have cancer, then face a recall, and often a biopsy — a needle, a wait, a scar — only to be told they were fine all along. Screening also finds some cancers that would never have caused harm in a person's lifetime, and treats them anyway: surgery and radiotherapy for a disease that was never going to bite. That's the quiet cost economists call overdiagnosis.

Against that sits the obvious benefit: real cancers caught early, when they're most treatable. Screening genuinely saves lives. The hard question is never "does it work" but "for whom does the good outweigh the harm" — and the answer turns entirely on the base rate. Breast cancer is rare at forty and steadily less rare at seventy. Feed a rarer disease into the same test and the false alarms swell relative to the real finds. That single fact is why guidelines argue over the starting age, and why the same test can be sound policy at sixty and a closer call at forty.

06 So what?

The lesson reaches far past mammograms. Any test for a rare thing — a spam filter, an airport scanner, a sniffer dog, a fraud alert, a rare-disease screen — lives under the same arithmetic. When the thing you're hunting is scarce, even a very accurate test throws off more false alarms than true finds, and a single positive is the start of an inquiry, not the end of one. Ask not just "how accurate is the test" but "how rare is the thing", and always, always ask what happens next to the people who test positive.

A positive result is not a diagnosis. It's a nudge — and how big a nudge depends entirely on how rare the thing was to begin with.