THE LOGIC OF CHANCE.* Tins is a very able and, to the present reviewer at least, a very entertaining little book on thelogic of chance, though it does not strike us as quite so original as Mr. Veun thinks it, and we do not agree with the general theory it contains. The main subject of the book is the old controversy as to what we mean by the chance of an event,—whether probability is a property of thoughts or things,—whether it belongs to individual events as individual events, like single throws of a die, or only expresses the pro- portions between the various results of events of the same kind, sires, fives, fours, &c., when thrown in a considerable series, — whether it is a science founded solely on adequate statistical results, that is, on knowledge of proportions between different classes of events, as, for example, the ratio of male to female births, —or whether it can ever be applied, in our ignorance o'f the real proportions, to show us the direction in which such little know- ledge as we have gained clearly tends. On almost all these points we differ from Mr. Venn, though entirely agreeing with him as to the absurdity of applying the principles of probability in any case in which the slightest item of moral evidence at our disposal entirely outweighs all the value of the numerical laws which it is possible for us to apply,—of determining, for instance, the chance of a soldier's death from any particular wound by the number of fatal cases in general, 'to the exclusion of the far more important particular symptoms in the individual patient's case.
First, as to Mr. Venn's assertion that probability is properly a property of things, not of our own conceptions of things. With this we should agree, if only he admits our own conceptions amongst the things which have the property of probability in question. There are passages in his book which seem to object to founding any inference on our own ignorance. Now, supposing I happen to know that of two turnings one will lead me where I want to go, and one will not, but don't know and have no suspicion which of the two is the right one, surely on this ignorance I can yet found so much knowledge as this,—that if in every case where I am so circumstanced I choose at random, i. e., do not choose at all, but let myself be determined by a purely accidental cir- cumstance which has no relation to rightness or wrongness, I shall in the long run go right exactly as often as I go wrong. And will not this be a piece of knowledge founded on my experience of absolute ignorance ? It may seem a very useless piece of knowledge founded upon ignorance, and yet it is not wholly so. For supposing there were two sets of ways of going to the place I am seeking, one of which goes by a turtling where three or four roads meet, and the other by the way mentioned before where only two roads meet, and I am equally ignorant which way to choose at both turnings, it is clear that I shall be wiser to follow the path where only two roads meet, for there my chances of error will be fewer than where three or four meet ; and so if my ignorance of two processes for doing the same thing is equally complete, but in one of them there is but one dangerous blunder possible, and in another two or three, I can estimate exactly the superior wisdom of choosing the less complex process. If this is true about the ramifications of things, it must be equally true about the ramifications of thoughts. I may clearly found a presumption as to truth or error on the same principles on which I found a presumption as to the right way or the wrong. The true answer to a question may be known to others, and yet if I do not know it and cannot know it, I may be quite right in treating my presumption as to the correct answer as a probability, and estimating the chance of its being right, exactly as I would the chance of a die turning up a particular face. Thoughts are things for all the purposes of this science, and there are cases in which it is quite easy to assign an arithmetical value to the probability of their turning out correct thoughts, where we cannot have access to the facts necessary to determine whether they are correct or not. If we can study the facts, we try to learn the possible limits of variation of the facts, though we do not know what causes the variation between those limits, as between the head and tail in a throw of a halfpenny. If we cannot study the facts, we try to determine the limits of variation of conceivably true guesses about the facts, though we do not know any reason for preferring one guess to the other. It is clear that the guess about the fact has just as much an assignable probability, as the fact itself.
The Logic of Chance. An Essay on the foundations and provioce of the theory of probability, with especial reference to its application to moral and social science. By John Venn, M.A., Fellow of Gonville and Caine College, Cambridge. London : Macmillan. Again, we cannot agree with Mr. Venn in considering that to apply the word probability strictly to an individual fact is a mistake, probability meaning, according to him, the proportion of events of one kind, say female births, to the whole in a sufficiently large series of more general events (human births). Thus Mr. Veun thinks that to say that the probability of a particular throw of a penny turning up heads' is one-half, only means that in the long run out of every hundred throws half will be heads and half tails. And he takes a good deal of pains to show that the it priori proof, as it is called,—that in any one case there is an equal tendency beforehand to head' and tail,'—is not really is priori, but rests upon the experience that in fact of all the actual throws made "with fair halfpennies," half have been in one way and half the other. Now, we quite admit that in the case of the halfpenny the a priori proof is a little doubtful. It is undoubtedly conceivable that if the same thrower always threw up the halfpenny with the head uppermost, there might be a predominance in one kind of result, and with another thrower a predominance of the opposite kind, so that we may really require the test of experience in this case to prove that practically the conditions are equally favourable to either result. But that the probability of various individual events can be assigned without any large number of trials, and even without one trial, seems to us. certain. Mark out the floor of a room with rows of squares on which the regular series of ordinary numbers are written, and then tell a blind man who has never heard of the arrangement to halt at random on any of the squares in succession. It is perfectly cer- tain, before any trial, that in the long run in wandering about the room he will halt as often on even numbers as on odd, for, not knowing of the arrangement, his reasons or caprices for halting must be independent of the evenness or oddness, and cannot favour either. And, therefore, we say that his chance of halting on an even number or on an odd is equal, without asking for statistical results. And what do we mean by calling the chance of the individual event for either case one-half ? Simply that the conditions tending to either event are of the same force, and that those which actually determine it in any individual case are accidental with regard to the issue, that is, as likely to determine it to the one issue as to the other. When Mr. Venu says that there is no such thing at all as a probability for an individual event, he seems to us to deny practically that there is such a thing as the necessity of an indi- vidual event or the impossibility of it. We say that an eclipse of the sun by the moon is necessary, if neither moon nor sun perish, or are drawn out of their orbits before the moment when the moon intervenes between us and the sun ; that it is impossible, if the moon does not intervene between us and the sun. Where an individual event can be necessary or impossible, it can surely be something between the two—probable. In other words, the whole causes tend- ing to produce it can bearan assignable ratio to the whole causes tend- ing either to produce or prevent it. If it is necessary that an eclipse shall happen if the forces at work bring the moon between the sun and us, it would be probable if some forces tended to bring it between the sun and us, and some tended to draw it away, while the issue depended on the unknown effect of the force to be exerted by- some variable attraction. The apparent contradiction of which Mr. Venn gets rid by denying probability' to individual events, is the contradiction of first postulating the event as an entity, and then suggesting a doubt as to whether it will ever really be one. But almost a similar objection might be made as to speaking of a future- event which cannot be an event until it is past. The truth is that all we mean by a probable event is, a conceived event the existing and calculable amount of tendency to the production of which is- greater than the similar existing and calculable tendency to. prevent its production. It is no doubt quite true, as Mr. Venn points out, that we can only judge of this tendency at all in many cases by statistical results. The science of their evolution is too much a secret, depends on too many complex causes, to be tested by individual analysis at all,—as in the case of the probable duration of human life. To calculate this for an in- dividual case except by arguing from a great number of cases is impossible. And most cases of probability are of like kind. But because there are two modes of measuring probability,—the intensive, which measures the tendency in the individual case,— and the extensive, which gauges the general amount of causes acting upon the collective class of cases,—we have no reason to deny the former, which, when it can be calculated at all, is far the more accurate. The probability which we measure by tendency before the event,—as in the case of a halfpenny or a die,—seems to us to bear to the probability which we measure by statistics after a good number of such events, the same relation which the formula from which a series is evolved, bears to the developed series. We may obtain the latter directly by experiment, but if we know the formula which produced it, it is always far more convenient to deal with that. Mr. Venn sees that you can obtain it a priori in so few cases that are of practical value, that he wants to deny himself the advantage of it in those few cases which have really led to the mathematical theory of probability.
Many of Mr. Venn's remarks on the attempt to apply the laws of probability to cases on which we have far more moral evidence that we cannot gauge by quantity at all—than any mathematical rules will supply us, are very good. It is quite true that it is absurd to apply the laws of probability derived from the ideal case of drawing balls from a box, to solving such a question as what we ought to expect when ten ships have sailed past us with a flag, with respect to the eleventh also having a flag or the same flag. The conjectures we are able to form as to the fleet in question, would usually be far more important than the principle which we can deduce from the fact that we have drawn ten balls of the same colour from a box, as to our legitimate expectation concerning the eleventh drawing. But Mr. Venn does not state fairly the law of ' inverse probability,' when he says that it professes to assign a different answer to the following questions. (1.) What is the chance that if A B dies he will die of typhus fever ? (2.) A B is dead. What is the chance that he died of typhus? The rule of inverse probability really does help us to detect the most probable instantaneous law in operation from our knowledge of a few specific instances. But it is applicable only where we know that there are various laws equally probable in themselves, but not equally likely to yield the actual event. A fair illustration would have been this. Given a class of diseases equally prevalent, and from which men are about equally likely to suffer, but verydifferent in their fatality, of which typhus is one, and then ask (1), what is the chance that, if A B takes one of these diseases (say typhus), he will die ? (2.) He is dead from one of these diseases ; what is the chance he died of typhus ? The second ques- tion would be quite distinct from the first, and would be a question soluble only by the principle of inverse probability. No doubt Mr. Venn will say that such questions -are not very useful, and that the condition they involve,—that all the supposed laws shall be equally likely in themselves, though not equally likely to produce the given event,—robs the principle of much of its practical value. No doubt that is so, but in at least one case,—the deter- mination of the law of error for the observations of astronomy, the law shown to be most likely by theory has also proved practically of the highest utility, and here at least Mr. Venn should allow, on his own principle, that experience has borne us out in the bold assumption, which it priori we admit to have been quite uncertain, namely, that of all the possible chances assumed, any one is intrinsically as likely as any other.
Little as we agree with many of the theoretic conclusions arrived at by Mr. Venn, it is impossible to read his very clever book without both amusement and instruction.