Don't worry your life away

This book has one idea to impart, but it's an important one. Written by a professor of psychology in Germany, it offers a technique - a "mind tool", the author calls it - that makes clear thinking easier in one particular family of medical and legal situations, situations where it's vital to be able to gauge the degree of certainty with which you're being told something.

Suppose you're signed up in a breast cancer screening programme (the situation Gerd Gigerenzer takes as his central example). Horror of horrors, your mammogram comes back positive. But you shouldn't panic.

Death and taxes may be certain, says Gigerenzer, following Benjamin Franklin, but few other things are, and certainly not medical tests, most of which are subject to a version of Sod's Law under which the better they are at detecting a condition that's actually there, the more often they'll detect it when it isn't. So how likely is it that you really have breast cancer? To answer this question, you're going to have to embrace the fact that the world is governed by probabilities rather than certainties. You'll have to reason statistically; which is doubly hard, because not only do most patients find statistics impenetrable, so do most doctors. But Gigerenzer knows how to make it easier.

You need, he explains, to know three things. One, the probability of breast cancer among women your age. Two, the probability of the test detecting it when it's there. Three, the probability of the test detecting it when it isn't there.

The conventional method expresses these three bits of data as percentages: the genuine figures for women aged between 40 and 50, for example, are 0.8 per cent, 90 per cent and seven per cent, respectively. Then it plugs them into a useful equation invented by an 18th century clergyman called Thomas Bayes, to extract the probability that in your case, the test result is true.

Unfortunately, most of us lose the thread about here. We lack the ability to think things through quickly and naturally in this abstract way; we're "innumerate". Gigerenzer, though, believes that a great deal of apparent innumeracy is caused by data being presented to us in ways that don't "match our minds' design". We can abolish the confusion by using "natural frequencies" instead of probabilities. In other words, you should think of your three bits of breast-cancer data in terms of the concrete numbers of women they represent, not abstract that of those eight cases of breast cancer, the test catches seven. So seven women with breast cancer will test positive out of 1,000.

But the test also detects it in seven per cent of cases when it isn't there, or in 70 women out of 1,000. That's 70 "false positives". The total number of positives is seven plus 70 equals 77, out of which only seven are true positives.

So what's your chance of really having breast cancer if you test positive? Only seven chances in 77, or one in 11. There's still an appreciable risk of cancer here, but no one should decide on the basis of one positive mammogram that cancer is inevitable, or even likely. Crucially, this is still Thomas Bayes's equation at work, only put into a form where it becomes commonsensical.

Gigerenzer goes on to spell out the implications for breast cancer screening, which are counterintuitive, and worrying. The vast majority of women who undergo biopsies or exploratory surgery on the basis of a mammogram do not have breast cancer: 100,000 a year in Germany. If you have no symptoms, and are under 50, screening does not reduce your chance of dying of breast cancer. (The fashion industry's breast cancer campaign, with its central focus on women in their teens and twenties, looks really silly in this light: just another way for the purveyors of size eight clothing to make people anxious.) Nor is the breast cancer test the only one to be structured this way.

The problem with false positives is built in all over the place. Gigerenzer leads you through the steps required to allow for them in HIV tests, in courtroom DNA matching and in claims about violent psychiatric patients.