Chance or Probability

The aim of science is to describe the world in orderly language, in such a way that we can, if possible, foresee the results of those alternative courses of action between which we are always choosing. The kind of order which our description has is entirely one of convenience. Our purpose is always to predict. Of course, it is most convenient if we can find an order by cause and effect; it makes our choice simple; but it is not essential.

There is of course nothing sacred about the causal form of natural laws. We are accustomed to this form, until it has become our standard of what every natural law ought to look like. If you halve the space which a gas fills, and keep other things constant, then you will double the pressure, we say. If you do such and such, the result will be so and so; and it will always be so and so. And we feel by long habit that it is this 'always' which turns the prediction into a law. But of course there is no reason why laws should have this always, all-or-nothing form. If you self-cross the offspring of a pure white and a pure pink sweet pea, said Mendel, then on an average one-quarter of these grandchildren will be white, and three-quarters will be pink. This is as good a law as any other; it says what will happen, in good quantitative terms, and what it says turns out to be true. It is not any less respectable for not making that parade of every time certainty which the law of gases makes.

It is important to seize this point. If I say that after a fine week, it always rains on Sunday, then this is recognized and respected as law. But if I say that after a fine week, it rains on Sunday more often than not, then this somehow is felt to be an unsatisfactory statement; and it is taken for granted that I have not really got down to some underlying law which would chime with our habit of wanting science to say decisively either 'always' or 'never'. Somehow it seems to lack the force of law.

Yet this is a mere prejudice. It is nice to have laws which say, 'This configuration of facts will always be followed by event A, ten times out of ten.' But neither taste nor convenience really make this a more essential form of law than one which says, 'This configuration of facts will be followed by event A seven times out of ten, and by event B three times out of ten.' In form the first is a causal law and the second a statistical law. But in content and in application, there is no reason to prefer one to the other.

There is, however, a limitation within every law which does not contain the word 'always'. Bluntly, when I say that a configuration of facts will be followed sometimes by event A and at other times by B, I cannot be certain whether at the next trial A or B will turn up. I may know that A is to turn up seven times and B three times out of ten; but that brings me no nearer at all to knowing which is to turn up on the one occasion I have my eye on next time. Mendel's law is all very fine when you grow sweet peas by the acre; but it does not tell you, and cannot, whether the single second generation seed in your window box will flower white or pink.

But this limitation carries with it a less obvious one. If we are not sure whether A or B will turn up next time, then neither can we be sure which will turn up the time after, or the time after that. We know that A is to turn up seven times and B three; but this can never mean that every set of ten trials will give us exactly seven As and three Bs.

Then what do I mean by saying that we expect A to turn up seven times to every three times which B turns up? I mean that among all the sets of ten trials which we can choose from an extended series, picking as we like, the greatest number will contain seven As and three Bs. This is the same thing as saying that if we have enough trials, the proportion of As to Bs will tend to the ratio of seven to three. But of course, no run of trials, however extended, is necessarily long enough. In no run of trials can we be sure of reaching precisely the balance of seven to three.

Then how do I know that the law is in fact seven As and three Bs? What do I mean by saying that the ratio tends to this in a long trial, when I never know if the trial is long enough? And more, when I know that at the very moment when we have reached precisely this ratio, the next single trial must upset it because it must add either a whole A or a whole B, and cannot add seven-tenths of one and three-tenths of the other. I mean this. After ten trials, we may have eight As and only two Bs; it is not at all improbable. But it is very improbable that, after a hundred trials, we shall have as many as eighty As. It is excessively improbable that after a thousand trials we shall have as many as eight hundred As; indeed it is highly improbable that at this stage the ratio of As and Bs departs from seven to three by as much as five per cent. And if after a hundred thousand trials we should get a ratio which differs from our law by as much as one per cent, then we should have to face the fact that the law itself is almost certainly in error.

Let me quote a practical example. The great naturalist Bufhon was a man of wide interests. His interest in the laws of chance prompted him to ask an interesting question. If a needle is thrown at random on a sheet of paper ruled with lines whose distance apart is exactly equal to the length of the needle, how often can it be expected to fall on a line and how often into a blank space? The answer is rather odd: it should fall on a line a little less than two times out of three - precisely, it should fall on a line two times out of pi where pi is the familiar ratio of the circumference of a circle to its diameter, which has the value 3.14159265.... How near can we get to this answer in actual trials? This depends of course on the care with which we rule the lines and do the throwing; but, after that, it depends only on our patience. In 1901 an Italian mathematician, having taken due care, demonstrated his patience by making well over 3,000 throws. The value he got for pi was right to the sixth place of decimals, which is an error of only a hundred thousandth part of one per cent.

This is the method to which modern science is moving. It uses no principle but that of forecasting with as much assurance as possible, but with no more than is possible. That is, it idealizes the future from the outset, not as completely determined, but as determined within a defined area of uncertainty.

(From The Common Sense of Science by J. Bronowski.)