“But is this the full truth of the matter? Are not the laws of chance a source of knowledge? And, stranger still, is it not sometimes easier to generalise (say) about random movements than about movements which obey even a simple law—witness the kinetic theory of gases? And, if this be so, how can chance be the equivalent of ignorance? Ask a physicist to explain what goes on in a gas. He might, perhaps, express his views in some such terms as these: ‘You wish me to tell you about these complex phenomena. If by ill luck I happened to know the laws which govern them, I should be helpless. I should be lost in endless calculations, and could never hope to supply you with an answer to your questions. Fortunately for both of us, I am completely ignorant about the matter; I can, therefore, supply you with an answer at once. This may seem odd. But there is something odder still, namely, that my answer will be right.’”
Now, what are the conditions which make it possible thus to extract a correct answer from material apparently so unpromising? They would seem to be a special combination of ignorance and knowledge, the joint effect of which is to justify us in supposing that the particular collection of facts or events with which we are concerned are happening “at random.” If we could calculate the complex causes which determine the fall of a penny, or the collisions of a molecule, we might conceivably deal with pennies or molecules individually; and the calculus of probability might be dispensed with. But we cannot; ignorance, therefore, real or assumed, is thus one of the conditions required to provide us with the kind of chaos to which the doctrine of chances may most fittingly be applied. But there is another condition not less needful, namely, knowledge—the knowledge that no extraneous cause or internal tendency is infecting our chaotic group with some bias or drift whereby its required randomness would be destroyed. Our penny must be symmetrical, and Maxwell’s demons[10] must not meddle with the molecules.
The slow disintegration of radium admirably illustrates the behaviour of a group or collection possessing all the qualities which we require. The myriad atoms of which the minutest visible fragment is composed are numerous enough to neutralise eccentricities such as those which, in the case of a game of chance, we call “runs of luck.” Of these atoms we have no individual knowledge. What we know of one we know of all; and we treat them not only as a collection, but as a collection made at random. Now, physicists tell us that out of any such random collection a certain proportion will disintegrate in a given time; and always the same proportion. But whence comes their confidence in the permanence of this ratio? Why are they so assured of its fixity that these random explosions are thought to provide us with a better time-keeper than the astronomical changes which have served mankind in that capacity through immemorial ages? The reason is that we have here the necessary ignorance and the necessary knowledge in a very complete form. Nothing can well exceed our ignorance of the differences between one individual radium atom and another, though relevant differences there must be. Nothing, again, seems better assured than our knowledge that no special bias or drift will make one collection of these atoms behave differently from another. For the atomic disintegration is due to no external shock or mutual reaction which might affect not one atom only, but the whole group. A milligram of radium is not like a magazine of shells, where if one spontaneously explodes all the rest follow suit. The disruption of the atom is due to some internal principle of decay whose effects no known external agent can either hasten or retard. Although, therefore, the proportion of atoms which will disintegrate in a given time can only be discovered, like the annual death-rate among men, by observation, yet once discovered it is discovered for ever. Our human death-rate not only may change, but does change. The death-rate of radium atoms changes not. In the one case, causes are in operation which modify both the organism and the surroundings on which its life depends. In the other case, it would seem that the average of successive generations of atoms does not vary, and that, once brought into existence, they severally run their appointed course unaffected by each other or by the world outside.
So far we have been concerned with groups or collections or series; and about these the doctrine of chances and the theory of error may apparently supply most valuable information. But in practical affairs—nay, even in many questions of scientific speculation—we are yet more concerned about individual happenings. We have, therefore, next to ask how we can infer the probability of a particular event from our knowledge of some group or series to which it belongs.
There seems at first sight no difficulty in this, provided we have sufficient knowledge of the group or series of which the particular event is a member. If we know that a tossed penny will in the long run give heads and tails equally often, we do not hesitate to declare that the chances of a particular throw giving “heads” are even. To expect in any given case heads rather than tails, or tails rather than heads, is inconsistent with the objective knowledge of the series which by hypothesis we actually possess.
But what if our information about the group or series is much less than this? Suppose that, instead of knowing that the two possible alternatives do in fact occur equally often, we are in the less advantageous position of knowing no reason why they should not occur equally often. We ought, I suppose, still to regard the chances of a particular toss as even; although this estimate, expressed by the same fraction (½) and held with the same confidence, is apparently a conclusion based on ignorance, whereas the first conclusion was apparently based on knowledge.
If, for example, we know that a die is fairly made and fairly thrown, we can tell how often a particular number will turn up in a long series of throws, and we can tell what the chances are that it will turn up on the occasion of a single throw. Moreover, the two conclusions seem to be logically connected.
But if we know that the die is loaded we can no longer say how the numbers will be distributed in a series of throws, however long, though we are sure that the distribution will be very different from what it would have been had the die been a fair one. Nevertheless, we can still say (before the event) what the chances are of a particular number turning up on a single throw; and these chances are exactly the same whether the die be loaded or whether it be fair—namely, one-sixth. Our objective knowledge of the group or series has vanished, but, with the theory of probability to help us, our subjective conviction on this point apparently remains unchanged.
There is here, surely, a rather awkward transition from the “objective” to the “subjective” point of view. We were dealing, in the first case, with groups or series of events about which the doctrine of chances enabled us to say something positive, something which experience would always confirm if the groups or series were large enough. A perfect calculator, endowed with complete knowledge of all the separate group members, would have no correction to make in our conclusions. His information would be more complete than our own, but not more accurate. It is true that for him “averages” would have no interest and “chance” no meaning. Nevertheless, he would agree that in a long series of fair throws of a fair die any selected face would turn up one-sixth times as often as all the others taken together. But in the second case this is no longer so. Foresight based on complete knowledge would apparently differ from foresight based on the calculation of chances. Our calculator would be aware of the exact manner in which the die was loaded, and of the exact advantage which this gave to certain numbers. He would, therefore, know that in asserting the chance of any particular number turning up on the first throw to be one-sixth, we were wrong. In what sense, then, do we deem ourselves to have been right?
The answer, I suppose, is that we were right not about a group of throws made with this loaded die, but about a group of such groups made with dice loaded at random—a group in which “randomness” was so happily preserved among its constituent groups that its absence within each of these groups was immaterial, and no one of the six alternative numbers was favoured above another.