This article was taken from the September 2011 issue of Wired magazine. Be the first to read Wired's articles in print before they're posted online, and get your hands on loads of additional content by subscribing online.
Why do so many people find probability theory so unintuitive and difficult? After years of careful study, I have finally found it's because probability is unintuitive and difficult.
Consider relative and absolute risks. A bacon sandwich a day increases your risk of bowel cancer by 20 percent. This sounds frightening: but if we think of 100 people stuffing their faces each day with a greasy sarnie, then this 20 percent increase means the five who we would expect to get bowel cancer anyway will go up to six. In other words just one in 100 will be harmed in this way.
I'm told that, if I take statins, my risk of a heart attack or stroke in the next ten years will go down by 30 per cent. So, out of 100 ways things may turn out for me in ten years' time, taking statins every day will reduce the number of heart attacks or strokes from ten to seven.
With only a three in 100 chance of benefiting, and possible side effects, I don't think I'll bother.
The use of relative risks -- 20 percent increase, 30 percent decrease and so on -- makes effects look bigger, and is an example of the framing of a risk. A recent story illustrates another trick.
A genetic variant present in ten percent of people was found by researchers to protect against high blood pressure. Although published in a top scientific journal, the story got negligible press coverage -- until a clever press officer rewrote the press release to say that a genetic variant had been found that increased the risk of high blood pressure in 90 percent of people.
A classic example concerns screening programmes. Mammography correctly classifies around 90 percent of women who go for breast-cancer screening. So when a middle-aged woman is told she has a positive test result, what's the probability she doesn't have cancer? The answer, which is surprising to most people, is around 91 per cent. The crucial missing piece of information is the size of her background risk. So suppose she is from a population in which around one in 100 have breast cancer. Then, out of 100 such women tested, one would have breast cancer and will most likely test positive. But of the 99 who do not have breast cancer, we would still expect around ten to test positive -- as the test is only 90 percent accurate. That makes 11 positive tests, only one of which involves cancer, which gives a 10/11 = 91 percent probability that someone who tests positive does not have breast cancer.
An unsurprising characteristic of media reporting is that it concentrates on things, usually nasty, that happen rather than things that don't. We search in vain for headlines such as "No children killed on the way to school today". If we think of a probability as being a numerator and a denominator (eg two out of 100), then narratives look only at numerators and so suffer from what is known as "denominator neglect", amplifying the apparent risk of an event. And the rarer an event becomes, the more coverage it gets when it does happen. Children are fortunately far safer than they have ever been, and so a child death is given great prominence. No passenger has been killed in a train accident in the UK since February 2007, and so there will be massive attention when a fatal accident does occur -- though it is notable that around 250 to 300 trespassers and suicides are killed by trains each year in the UK, a steady number that receives little coverage.
So if you read about a horrible event, and you have an interest in the risk of its happening again, ask, "How many opportunities were there for it to occur?"
David Spiegelhalter is Winton Professor of the Public Understanding of Risk, in the Statistical Lab at the University of Cambridge
This article was originally published by WIRED UK
