The Calculus Of Probabilities

What has that calculus to do with physical science?

However solidly founded a prediction may appear to be, we are never absolutely certain that experiment will not prove it false.

But the probability is often so great that practically it may be accepted.

Thus, in a multitude of circumstances, the physicist is often in the same position as the gambler who reckons up his chances.

Every time that he reasons by induction, he consciously requires the calculus of probabilities. That is why I am obliged:

• to open this chapter parenthetically, and
• to interrupt our discussion of method in the physical sciences in order to examine what this calculus is worth, and what dependence we may place on it.

The very name of the calculus of probabilities is a paradox.

Probability as opposed to certainty is what one does not know, and how can we calculate the unknown?

Yet many eminent scientists have devoted themselves to this calculus, and it cannot be denied that science has drawn therefrom no small advantage. How can we explain this apparent contradiction?

Has probability been defined? Can it even be defined?

If it cannot, how can we venture to reason upon it?

The definition, it will be said, is very simple.

The probability of an event is the ratio of the number of cases favourable to the event to the total number of pos- sible cases.

A simple example will show how incomplete this definition is:—I throw two dice.

What is the probability that one of the two at least turns up a 6?

Each can turn up in six different ways; the number of possi- ble cases is 6 × 6 = 36. The number of favourable cases 11 is 11; the probability is . That is the correct solution. 36 But why cannot we just as well proceed as follows?—The 206 6 × 7 points which turn up on the two dice form = 21 2 different combinations. Among these combinations, six 6 are favourable; the probability is . Now why is the 21 first method of calculating the number of possible cases more legitimate than the second? In any case it is not the definition that tells us.

We are therefore bound to complete the definition by saying, “. . . to the total number of possible cases, provided the cases are equally probable.”

So we are compelled to define the probable by the probable.

How can we know that two possible cases are equally probable?

Will it be by a convention?

If we insert at the beginning of every problem an explicit convention, well and good! We then have nothing to do but to apply the rules of arithmetic and algebra, and we complete our cal- culation, when our result cannot be called in question.

But if we wish to make the slightest application of this result, we must prove that our convention is legitimate, and we shall find ourselves in the presence of the very difficulty we thought we had avoided. It may be said that common-sense is enough to show us the convention that should be adopted.

Alas! M. Bertrand has amused himself by discussing the following simple problem:—“What is the probability that a chord of a circle may be greater than the side of the inscribed equilateral triangle?”

The illustrious geometer successively adopted two conventions which seemed to be equally imperative in the eyes of common-sense, and with one convention he finds 12 , and with the other 13 .

The conclusion which seems to follow from this is that the calculus of probabilities is a useless science, that the obscure instinct which we call common-sense, and to which we appeal for the legitimisation of our conventions, must be distrusted.

But to this conclusion we can no longer subscribe. We cannot do without that obscure instinct. Without it, science would be impossible, and without it we could neither discover nor apply a law. Have we any right, for instance, to enunciate Newton’s law?

No doubt numerous observations are in agreement with it, but is not that a simple fact of chance? and how do we know, besides, that this law which has been true for so many generations will not be untrue in the next? To this objection the only answer you can give is: It is very improbable. But grant the law.

By means of it I can calculate the position of Jupiter in a year from now. Yet have I any right to say this?

Who can tell if a gigantic mass of enormous velocity is not going to pass near the solar system and produce unforeseen perturbations?

Here again the only answer is: It is very improbable.

From this point of view all the sciences would only be unconscious applications of the calculus of probabilities. And if this calculus be condemned, then the whole of the sciences must also be condemned.

I shall not dwell at length on scientific problems in which the intervention of the calculus of probabilities is more evident.

In the forefront of these is the problem of interpolation, in which, knowing a certain number of values of a function, we try to discover the intermediary values.

I may also mention the celebrated theory of errors of observation, to which I shall return later;

The kinetic theory of gases is a well-known hypothesis wherein each gaseous molecule is supposed to describe an extremely complicated path, but in which, through the effect of great numbers, the mean phenomena which are all we observe obey the simple laws of Mariotte and Gay-Lussac.

All these theories are based upon the laws of great numbers, and the calculus of probabilities would evidently involve them in its ruin. It is true that they have only a particular interest, and that, save as far as interpolation is concerned, they are sacrifices to which we might readily be resigned.

But it would not be these partial sacrifices that would be in question; it would be the legitimacy of the whole of science that would be challenged. I quite see that it might be said: We do not know, and yet we must act. As for action, we have not time to devote ourselves to an inquiry that will suffice to dispel our ignorance.

Besides, such an inquiry would demand unlimited time.

We must therefore make up our minds without knowing.

This must be often done whatever may happen, and we must follow the rules although we may have but little confidence in them.

What I know is, not that such a thing is true, but that the best course for me is to act as if it were true.

The calculus of probabilities, and therefore science itself, would be no longer of any practical value.

Unfortunately the difficulty does not thus disappear.

A gambler wants to try a coup, and he asks my advice.

If I give it him, I use the calculus of probabilities; but I shall not guarantee success.

That is what I shall call subjective probability.

In this case we might be content with the explanation of which I have just given a sketch.

But assume that an observer is present at the play, that he knows of the coup, and that play goes on for a long time, and that he makes a summary of his notes.

He will find that events have taken place in conformity with the laws of the calculus of probabilities. That is what I shall call objective probability, and it is this phenomenon which has to be explained.

There are numerous Insurance Societies which apply the rules of the calculus of probabilities, and they distribute to their shareholders dividends, the objective reality of which cannot be contested. In order to explain them, we must do more than invoke our ignorance and the necessity of action.

Thus, absolute scepticism is not admissible. We may distrust, but we cannot condemn en bloc. Discussion is necessary.

I. Classification of the Problems of Probability

In order to classify the problems which are presented to us with reference to probabilities, we must look at them from different points of view, and first of all, from that of generality. I said above that probability is the ratio of the number of favourable to the number of possible cases. What for want of a better term I call generality will increase with the number of possible cases. This number may be finite, as, for instance, if we take a throw of the dice in which the number of possible cases is 36.

That is the first degree of generality. But if we ask, for instance, what is the probability that a point within a circle is within the inscribed square, there are as many possible cases as there are points in the circle—that is to say, an infinite number. This is the second degree of generality. Generality can be pushed further still. We may ask the probability that a function will satisfy a given condition.

There are then as many possible cases as one can imagine different functions. This is the third degree of generality, which we reach, for instance, when we try to find the most probable law after a finite number of observations.

Yet we may place ourselves at a quite different point of view. If we were not ignorant there would be no probability, there could only be certainty. But our ignorance cannot be absolute, for then there would be no longer any probability at all.

Thus the problems of probability may be classed according to the greater or less depth of this ignorance. In mathematics we may set ourselves problems in probability. What is the probabil- ity that the fifth decimal of a logarithm taken at random from a table is a 9.

There is no hesitation in answering 1. Here we possess all the data that this probability is 10 of the problem.

We can calculate our logarithm without having recourse to the table, but we need not give our- selves the trouble. This is the first degree of ignorance. In the physical sciences our ignorance is already greater. The state of a system at a given moment depends on two things—its initial state, and the law according to which that state varies. If we know both this law and this initial state, we have a simple mathematical problem to solve, and we fall back upon our first degree of ignorance. Then it often happens that we know the law and do not know the initial state.