Page:Encyclopædia Britannica, Ninth Edition, v. 19.djvu/793

Rh PROBABILITY 769 theoretical problem. Thus, it may be that a person accords a lower degree of credence to a fact attested by two or more independent witnesses than theory warrants, the reason being that he has unconsciously recognized the possibility of collusion, which had not been presented among the data. Again, it appears from the rules for the credibility of testimony that the probability of a fact may be diminished by being attested by a new witness, viz., in the case where his credibility is less than |. This is certainly at variance with our natural impression, which is that our previous conviction of any fact is clearly not weakened, however little it be intensified, by any fresh evidence, however suspicious, as to its truth. But on reflexion we see that it is a practical absurdity to suppose the credibility of any witness less than that is, that he speaks falsehood oftener than truth for all men tell the truth probably nine times out of ten, and only deviate from it when their passions or interests are concerned. Even where his interests are at stake, no man has any preference for a lie, as such, above the truth ; so that his testimony to a fact will at worst leave the antecedent probability exactly what it was. A celebrated instance of the confirmation and comple tion by theory of the ordinary view is afforded by what is known as James Bernoulli s theorem. If we know the odds in favour of an event to be three to two, as for instance that of drawing a white ball from a bag contain ing three white and two black, we should certainly judge that if we make five trials we are more likely to draw white three times and black twice than any other combination. Still, however, we should feel that this was very uncertain ; instead of three white, we might draw white 0, 1, 2, 4, or 5 times. But if we make say one thousand trials, we should feel confident that, although the numbers of white and black might not be in the proportion of three to two, they would be very nearly in that proportion. And the more the trials are multiplied the more closely would this proportion be found to obtain. This is the principle upon which we are continually judg ing of the possibility of events from what is observed in a certain number of cases. 1 Thus if, out of ten particular infants, six are found to live to the age of twenty, we judge, but with a very low amount of conviction, that nearly six-tenths of the whole number born live to twenty. But if, out of 1,000,000 cases, we find that 600,000 live to be twenty, we should feel certain that the same propor tion would be found to hold almost exactly were it possible to test the whole number of cases, say in England during the 19th century. In fact we may say, considering how seldom we know a priori the probability of any event, that the knowledge we have of such probability in any case is en tirely derived from this principle, viz., that the proportion which holds in a large number of trials will be found to hold in the total number, even when this may be infinite, the deviation or error being less and less as the trials are multiplied. Such no doubt is the verdict of the common sense of mankind, and it is not easy to say upon what considera tions it is based, if it be not the effect of the unconscious habit which all men acquire of weighing chances and probabilities, in the state of ignorance and uncertainty which human life is. It is now extremely interesting to see the results of the unerring methods of mathematical analysis when applied to the same problem. It is a very 1 So it is said, &quot;the tree is known by its fruits&quot;; &quot; practice is better than theory&quot; ; and the universal sense of mankind judges that the safest test of any new invention, system, or institution is to see how it works. So little are we able by a priori speculations to forecast the thousand obstacles and disturbing influences which manifest them selves when any new cause or agent is introduced as a factor in the world s affairs. difficult one, and James Bernoulli tells us he reflected upon it for twenty years. His methods, extended by De Moivre and Laplace, fully confirm the conclusions of rough common sense ; but they have done much more. They enable us to estimate exactly how far we can rely on the proportion of cases in a large number of trials, truly representing the proportion out of the total number that is, the real probability of the event. Thus he proves that if, as in the case above mentioned, the real probability of an event is f, the odds are 1000 to 1 that, in 25,550 trials, the event shall occur not more than 15,841 times and not less than 14,819 times, that is, that the deviation from 15,330, or f of the whole, shall not exceed -^ of the whole number of trials. The history of the theery of probability, from the celebrated question as to the equitable division of the stakes between two players on their game being inter rupted, proposed to Pascal by the Chevalier de Mere in 1654, embracing, as it does, contributions from almost all the great names of Europe during the period, down to Laplace and Poisson, is elaborately and admirably given by Mr Todhunter in his History of the subject, now a classical work. It was not indeed to be anticipated that a new science which took its rise in games of chance, and which had long to encounter an obloquy, hardly yet extinct, due to the prevailing idea that its only end was to facilitate and encourage the calculations of gamblers, could ever have attained its present status that its aid should be called for in every department of natural science, both to assist in discovery, which it has repeatedly done (even in pure mathematics), to minimize the unavoid able errors of observation, and to detect the presence of causes as revealed by observed events. Nor are com mercial and other practical interests of life less indebted to it : 2 wherever the future has to be forecasted, risk to be provided against, or the true lessons to be deduced from statistics, it corrects for us the rough conjectures of common sense, and decides which course is really, accord ing to the lights of which we are in possession, the wisest for us to pursue. It is sui generis aiid unique as an application of mathematics, the only one, apparently, lying quite outside the field of physical science. De Moivre has remarked that, &quot; some of the problems about chance having a great appearance of simplicity, the mind is easily drawn into a belief that their solution may be attained by the mere strength of natural good sense&quot;; and it is with sur prise we find that they involve in many cases the most siibtle and difficult mathematical questions. It has been found to tax to the utmost the resources of analysis and the powers of invention of those who. have had to deal with the new cases and combinations which it has pre sented. Great, however, as are the strictly mathematical difficulties, they cannot be said to be the principal. Especially in the practical applications, to detach the problem from its surroundings in rerum natura, discard ing what is non-essential, rightly to estimate the extent of our knowledge respecting it, neither tacitly assuming as known what is not known, nor tacitly overlooking some datum, perhaps from its very obviousness, to make sure that events we are taking as independent are not really connected, or probably so, such are the preliminaries necessary before the question is put in the scientific form to which calculation can be applied, and failing which the - Men were surprised to hear that not only births, deaths, and mar riages, but the decisions of tribunals, the results of popular elections, the influence of punishments in checking crime, the comparative values of medical remedies, the probable limits of error in numerical results in every department of physical inquiry, the detection of causes, physical, social, and moral, nay, even the weight of evidence and the validity of logical argument, might come to be surveyed with the lynx-eyed scrutiny of a dispassionate analysis. Sir J. Herschel. XIX. --97