Page:Encyclopædia Britannica, Ninth Edition, v. 19.djvu/792

768 THE mathematical theory of probability is a science which aims at reducing to calculation, where possible, the amount of credence due to propositions or statements, or to the occurrence of events, future or past, more especi ally as contingent or dependent upon other propositions or events the probability of which is known. Any statement or (supposed) fact commands a certain amount of credence, varying from zero, which means con viction of its falsity, to absolute certainty, denoted by unity. An even chance, or the probability of an event which is as likely as not to happen, is represented by the fraction i. It is to be observed that will be the probability of an event about which we have no knowledge whatever, because if we can see that it is more likely to happen than not, or less likely than not, we must be in possession of some information respecting it. It has been proposed to form a sort of thermometrical scale, to which to refer the strength of the conviction we have in any given case. Thus if the twenty-six letters of the alphabet have been shaken together in a bag, and one letter be drawn, we feel a very feeble expectation that A has been the one taken. If two letters be drawn, we have still very little confidence that A is one of them ; if three be drawn, it is somewhat stronger ; and so on, till at last, if twenty-six be drawn, we are certain of the event, that is, of A having been taken. Probability, which necessarily implies uncertainty, is a consequence of our ignorance. To an omniscient Being there can be none. Why, for instance, if we throw up a shilling, are we uncertain whether it will turn up head or tail ? Because the shilling passes, in the interval, through a series of states which our knowledge is unable to predict or to follow. If we knew the exact position and state of motion of the coin as it leaves our hand, the exact value of the final impulse it receives, the laws of its motion as affected by the resistance of the air and gravity, and finally the nature of the ground at the exact spot where it falls, and the laws regulating the collision between the two substances, we could predict as certainly the result of the toss as we can which letter of the alphabet will be drawn after twenty-five have been taken and examined. The probability, or amount of conviction accorded to any fact or statement, is thus essentially subjective, and varies with the degree of knowledge of the mind to which the fact is presented (it is often indeed also influenced by passion and prejudice, which act powerfully in warping the judgment), so that, as Laplace observes, it is affected partly by our ignorance partly by our knowledge. Thus, if the question were put, Is lead heavier than silver? some persons would think it is, but would not be surprised if they were wrong ; others would say it is lighter ; while to a worker in metals probability would be superseded by certainty. Again, to take Laplace s illustration, there are three urns A, B, C, one of which contains black balls, the other two white balls ; a ball is drawn from the urn C, and we want to know the probability that it shall be black. If we do not know which of the urns contains the black balls, there is only one favourable chance out of three, and the probability is said to be J. But if a person knows that the urn A contains white balls, to him the uncertainty is confined to the urns B and C, and therefore the proba bility of the same event is . Finally to one who had found that A and B both contained white balls, the probability is converted into certainty. In common language, an event is usually said to be likely or probable if it is more likely to happen than not, or when, in mathematical language, its probability exceeds i ; and it is said to be improbable or unlikely when its probability is less than . Not that this sense is always adhered to ; for, in such a phrase as " It is likely to thunder to-day," we do not mean that is more likely than not, but that in our opinion the chance of thunder is greater than usual ; again, " Such a horse is likely to win the Derby," simply means that he has the best chance, though according to the betting that chance may be only . Such unsteady and elliptical employment of words has of course to be abandoned and replaced by strict definition, at least mentally, when they are made the subjects of mathematical analysis. Certainty, or absolute conviction, also, as generally understood, is different from the mathematical sense of the word certainty. It is very difficult and often impossible, as is pointed out in the celebrated Grammar of Assent, to draw out the grounds on which the human mind in each case yields that con viction, or assent, which, according to Newman, admits of no degrees, and either is entire or is not at all. 1 If, when walking on the beach, we find the letters " Constantinople " traced on the sand, we should feel, not a strong impression, but absolute certainty, that they were characters not drawn at random, but by one acquainted with the word so spelt. Again, we are certain of our own death as a future event ; we are certain, too, that Great Britain is an island ; yet in all such cases it would be very difficult, even for a practised intellect, to present in logical form the evidence, which nevertheless has compelled the mind in each instance to concede the point. 2 Mathematical certainty, which means that the contrary proposition is inconceivable, is thus different, though not perhaps as regards the force of the mental conviction, from moral or practical certainty. It is questionable whether the former kind of certainty is not entirely hypothetical, and whether it is ever attainable in any of the affairs or events of the real world around us. The truth of no conclusion can rise above that of the premises, of no theorem above that of the data. That two and two make four is an incon trovertible truth ; but before applying even it to a concrete instance we have to be assured that there were really two in each constituent group; and we can hardly have mathematical certainty of this, as the strange freaks of memory, the tricks of conjurors, &c., have often made apparent. There is no more remarkable feature in the mathematical theory of probability than the manner in which it has been found to harmonize with, and justify, the conclusions to which mankind have been led, not by reasoning, but by instinct and experience, both of the individual and of the race. At the same time it has corrected, extended, and invested them with a definiteness and precision of which these crude, though sound, appreciations of common sense were till then devoid. Even in cases where the theoretical result appears to differ from the common-sense view, it often happens that the latter may, though perhaps unknown to the mind itself, have taken account of circumstances in the case omitted in the data of the 1 " There is a sort of leap which most men make from a high pro bability to absolute assurance . . . analogous to the sudden consilience, or springing into one, T>f the two images seen by binocular vision, when gradually brought within a certain proximity." Sir J. Herschel, in Edin. Review, July 1850. 2 Archbishop Whately s jcu d esprit, Historic Doults respecting Napoleon fionaparte, is a good illustration of the difficulties there may

be in proviny a conclusion the certainty of which is absolute.