My choice for a scientific principle that I would like to see universally understood is Bayes' theorem, named after the eighteenth-century British mathematician Thomas Bayes. This result is a very simple calculation in elementary probability, and is hardly worthy of the name 'theorem'. But the consequences, if everybody took Bayes' theorem seriously, would be profound.

To get an idea of what the theorem says, imagine the following situation. You have just been to hospital, and taken a test for a rare but serious disease. If you have the disease, then the test is guaranteed to detect it, but occasionally the test yields a false positive. More precisely, let us suppose that for one person in a thousand without the disease, the test indicates that they have it.

A few days later, you hear that the test has come out positive. How worried should you be? If you do not know Bayes' theorem, then you will probably reason as follows. If you have the disease, then the test will always give the correct answer; and if you do not have the disease, then the test will give the correct answer 99.9 per cent of the time. So the test has almost certainly given the correct answer, and you should be very worried indeed.

However, this reasoning is completely incorrect - a fact that, as many surveys have shown, most doctors do not understand at all. The correct answer is that you do not know how worried you should be, until you know how rare the disease is. Suppose, for instance, that one person in 10,000 has the disease. Then, there are two ways in which your positive result could have arisen. Either you have the disease, or you do not have the disease but you got a false positive.

The chances of the first are one in 10,000, and the chances of the second are approximately one in 1,000 - since almost everybody does not have the disease, and of those, one in 1,000 get a positive result. Thus, you are 10 times more likely to get a false positive than you are likely to have the disease. Once you have the positive result, your chances of having the disease are 1 in 11 - enough to be worrying, but nothing like the near certainty that it initially seemed.

Rather than say precisely what Bayes' theorem is, let me simply say that if you have understood the above example, then you have understood the theorem. Why do I regard the theorem as so important? Because all of us, whether we like it or not, have to make decisions based upon statistical evidence. Most of the time, it does not matter too much if we make them irrationally, but sometimes it matters a lot.

If doctors understood Bayes' theorem, then they would allocate their resources better, and lives would be saved. But deciding how to organise the health service is just one of many examples of important political decisions, that affect all of us. If politicians understood Bayes' theorem, then they could spend our taxes far more efficiently. If journalists understood Bayes' theorem, then they would not let politicians get away with fallacious statistical arguments. And if everybody understood Bayes' theorem, then politicians would be less tempted to advance those arguments in the first place. All our lives would be improved, as a result.

Timothy Gowers is author of Mathematics: A Very Short Introduction (buy this book from Amazon (UK) or Amazon (USA)). See his website.