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If then we are asked whether there is such a thing as chance in the world or not, we should say certainly not. Nor is it in the least implied in Probability, as we have grounded it, that there should be. All that we have implied is our ignorance about many of the detailed events, which is an extremely different thing from declaring that these events actually lack the regularity upon which any knowledge could be built. We fully believe that the turning up of one face rather than another of a die, is “caused ;" that is, that if we could precisely repeat the antecedents any number of times we should always obtain the same result. But the adjustments upon which this result depend are so excessively complicated and delicate, that all prevision is impossible. Hence our ignorance about the event beforehand is just as complete and absolute as if there really were none of that regularity which affords what we call a cause. Therefore we abandon as hopeless any attempt to determine the single event, and we fall back upon our knowledge of the average, or that statistical knowledge which has been described ; and we make what we can out of this, which, as will have been gathered, amounts really to a good deal.
The question whether people believe in the existence of chance is of course quite distinct. Some would settle the matter summarily by declaring the belief in causation a necessary belief. But we think that any one who had ever tried to ascertain what an ill-trained or puzzle-headed person does believe, will find that he has got quite enough to do there without attempting also to determine whether he believes it necessarily. The problem seems to us a hopeless one, and also one of no importance. Before we can get any one to say rationally whether he believes a doctrine, or can try to decide for him whether or not he does believe it, we must make him understand the doctrine. Otherwise, if we attempt to elicit an answer from his behavior we shall obtain contradictory replies. In certain respects he proceeds on one theory, in certain others on another, generally confounding the two together. It is quite enough, therefore, to remark that all cultivated minds seem to have entirely rejected the doctrine that any events occur casually; that is, without a regular phenomenal cause.
The true antithesis, then, is not one between what is casual
and what is causal ; it is rather between what is casual and what is designed. That is, we cannot draw a line between what is brought about by causes and what is not so brought about, for in the scientific view every event without exception has its cause. But, within this boundless range of causation and regularity, some events are brought about by the process which we may call “chance production,” and others are brought about by design. These two classes are not logical contradictories, in the sense that every event must belong to one or other of them; and in many cases it is extremely difficult to decide to which of them any particular event must be reserred. But there is probably no branch of the subject of Probability around which a greater quantity of popular philosophic interest has gathered, and therefore we must say something about it here.
We will begin with a simple example. Suppose that any one finds on a table, in a room ten coins, each with "head" uppermost; and the question is asked: “Is this result casual or de. signed?" Almost every one would say that it was designed;
" and if asked for his reason, would reply that it was very unlikely that the coins should all have turned up the same face accidentally. How unlikely it was can be settled in a moment ; ten coins, with two faces each, can be combined in 1024 ways, and only two of these give all the same face. Therefore once only in 512 times should we expect to find this regularity as the result of casual agency—that is, brought about in the way appropriate to the theory of chance. But then for a basis of comparison we ought to know how unlikely the other contingency is—that, namely, of design. The fact being undisputed that the coin has so turned up, the only doubt is as to the respective frequency of the two kinds of agency by which the event might have been brought to pass. It is here that we break down, owing to the hopeless inexactitude of all such problems. We do know how often a coin will turn up head of its own accord when duly tossed up, but we do not know how often human beings will turn it up so. Something depends on the kind of agents and the nature of their employment. Were the coins found in a gambling-house, we should conclude that, owing to the little time and taste the players probably had for
orderly arrangement of their coins, the designed arrangement was comparatively unlikely. On the other hand, had the table been that of a coin-collector, we might think it decidedly more likely that, for the purpose of comparison, he had set them all the same way uppermost. In this way we can do something towards indicating in what cases one event is more likely than others; but we can do nothing towards settling the problem in the only way in which the scientific man would regard it as settled, that is, towards saying how much more likely it is.
Is, then, all attempt at reasoning on this particular subject futile? By no means; if it were, an enormous amount of the decisions of practical life, to say nothing of those given in the law courts, would have to be rejected. Here, as in so many other cases, where we are quite unable to estimate differences numerically, we may still be able to assign a limit beyond which we feel safe in denying that the event can occur. A common laborer has never heard of a dynamometer, and has no conception of fixing how much stronger one of his fellow-workmen is than another, but he would risk his life on a bet that none of them could lift 2000 pounds. In some such way as this we are often enabled to bring the two classes of agencies—the definite numerical one which represents the pure chance, and the indefinite moral one which represents the design-into comparison with one another. Calculation may show that the former is so exceedingly small, that we feel quite confident that the latter, indeterminable as it is, must outweigh it. If 1000 coins were found, like the ten we mentioned, all face uppermost, no jury would believe a man on oath who declared that they had simply come so by a succession of fair tosses.
Let us now take a more complicated example. A Scotch astronomer, Mr. Piazzi Smyth, wrote a work entitled “Our Inheritance in the Great Pyramid." The work was intended to prove that a variety of modern scientific results were hidden away in different ways in that building; that its sides faced precisely to the points of the compass; that one of its passages pointed exactly to where the pole-star was situated a certain number of centuries ago, and so on. He also maintained that it contained a number of standard measures-mostly, as it happened, English ones; that its measurements were exactly expressible in English feet and inches; that a certain sarcophagus (as others had interpreted it) was really a standard quarter measure, and so forth. Among these results it was maintained that the length of the four sides, at the base, stood to the height in the exact proportion of the circumserence to the radius of a circle. We forget to how many decimals the result (of 3.14159, etc.) was guaranteed, but at any rate the precision attained was declared to be something far beyond the rude scientific knowledge of that age.
In such a case as this, even the assignment of the numerical chance becomes difficult, and can only be effected by the aid of various more or less arbitrary assumptions. In the first place we must set some limits to the height as compared with the base. If too high the building would be unsafe, if too low it would be ridiculous; consequently it is not from amongst all possible heights, but only from a limited range of them, that the selection could have been made. Then, in the next place, we must decide to how close an approximation the measurements have been made. If they are true to the hundredth part of an inch, the coincidence, if such it were, would have been much more astonishing. Suppose that this has been done, and that it has been ascertained that out of 10,000 possible heights for a pyramid of given base just that one has been selected which would give, in proportion to the circumserence of the base, the desired ratio of the radius to the circumference of the circle. We should then seem to be in possession of the elements which determine the “ chance” alternative, and when we had got hold of the elements of the “ design" alternative (that is, how likely builders were to employ the ratio in question, if they knew it, and how likely they were to know it) we might make the comparison. To attempt to give a numerical solution would be obviously futile, but, having raised the question, we may just remark that in our own judgment the coincidence was most likely not due to chance.'
* We must not conclude, however, with Mr. Smyth and his followers, that this would prove that the builders knew what was the ratio in question, but merely that they knew and employed some method which produced it. A teredo can bore a hole—that is, make a circle which exhibits its due ratios, as accurately as a geometrician; but we do not credit it with a knowledge of mathematics. An ingenious suggestion has been made by a writer in Nature, that the
In the above remarks we have assumed that, out of the 10,000 possibilities, only one ought to be regarded as favorable, viz., that which indicated the ratio in question of the circumference to the radius. But this is not warranted. The point is a somewhat subtle one, and we almost owe an apology for introducing it at the end of an article. But we cannot altogether pass it over, sor, as we have said, there probably is not a single side of the whole question of Probability which has given rise to an equal amount of that philosophical and theological discussion in which most of our readers will be likely to take an interest.
Suppose that in the above example the ratio had come out just double of that which it actually was, would not this have been taken as equal evidence of design? Or if it had proved to be double or treble the ratio of the diagonal of a square to one of its sides, would not this also have been of nearly equal significance? Proceeding in this way, we may suggest one known mathematical ratio after another until almost every single one of the 10,000 supposed possible intermediate positions has been occupied. If this were done, one might argue thus: Since every possible height of the pyramid would mark some mathematical value, a builder, ignorant of them all alike, could not help, nevertheless, stumbling upon one of them; why then attribute to him design in one case rather than in another? This shows that we have not got to the bottom of the question, and we had better, therefore, look again at some such simple problem as that of the ten coins. In this case it is readily seen that ten “heads” is just as likely, neither more nor less, than heads on seven specified coins, and tails on the other three.' Against each single specified arrangement the odds are the
builder may have proceeded somewhat as follows: Having decided on the height of his pyramid, he drew a circle with that as radius. Laying down a cord along the line of this circle, he then drew the cord out into a square, which square marked the base of the building. Hardly any simpler means could be devised in a comparatively rude age ; it is obvious that the circumference of the base, being equal to the length of the cord, would bear exactly the proper ratio to the height.
Seven specified coins. The chance that any seven should be head and the other three tail, is 120 times greater, being equal to tils or A.