8. Every event that can happen must happen, if trials enough be made ; and not only must happen, but must happen any number of times in succession. For example, if there be only one white ball to 100 black ones in an urn, and if drawings enough he made, the ball drawn being always returned and the urn properly shaken before a new trial, a person who goes on must at last draw that single white ball 1000 times running. This is a conclusion which the beginner cannot receive, particularly when he is told that 1,000,000 might be written for 1000, or any other number however high. Nevertheless, it will, in the course of his studies, not only be made clear to him, but it will also be shown that it is a conclusion which may be made obvious to common sense without any very profound mathematics.
9. If the odds against an event happening at any one trial be n to 1, there is an even chance of its happening in '69 x n trials,* it is 10 to 1 that it happens in 2'40 x n trials, 100 to 1 that it happens in 4.62 x n trials, 1000 to 1 that it happens in 6/1 x n trials, and 10,000 to 1 that it happens in 9/1 x n trials. Thus if it be 200 to 1 against success in any one trial, 9/1 x 200 being 1842, it is 10,000 to 1 that there shall be one success at least in about 1842 trials. But if a person should mako I000 trials without one success, ho would have no right to say that it is 10,000 to 1 in favour of one or more successes in the next 842 trials.
10. When any sum depends upon a contingent event, the present value of that sum is such a fraction of it as the probability of the event happening is of unity. Thus 201. to be gained if an event happen against which it is 7 to 1, is now worth only the eighth part of 20/.
There are no questions in the whole range of applied mathematics which require such close attention, and iu which it is so difficult to escape error, as those which occur in the theory of probabilities. This makes it hard to lay down in a abort space either maxims or examples sufficient to be really of use in advancing the student's progress ; and of all subjects, there is no one in which writers of every grade have so frequently or so strangely made mistakes of mere inadvertence. One was pointed out about twenty years ago (` Cam's. Phil. Trans.') into which both Laplace and Poisson had fallen, one after the other; but the discoverer of their slip proved himself signally liable to greater ones a very little while after. (` Cab. Cyclop. : Probability and Life p. 28.) We shall conclude by a brief account of the historical progress of this branch of science ' • referring the reader for more detail to lion tee's, and to the On Probability,' in the' Library of Useful Knowledge.' Those who cultivated games of chance must at all times have had a general notion of combinations which were more probable than others, and must have seen that those cases of which there were most to happen, always did iu reality happen most often. They could not fail to know, by reckoning on their fingers, that out of, for instance, all the throws of a pair of dice, there are only six doublets, and thirty other equally possible cases; nor could they have missed knowing that this must be the reason why doublets occur seldom in comparison with other throws. Notwithstanding this, the mathematical history of the
subject usually dates from a fragment by Galileo, which merely shows why 10 can be oftener thrown on three dice than 9, and two problems proposed by Chevalier do 3I6r6 to Pascal, in 1054, concerning certain points connected with games of chance.
That the history of correct investigation dates from this period there can be little doubt, but the subject had been previously considered by Cambia, in a work, ' De Ludo Alex,' published from his manuscript in the first volume of the collected edition of his works, and never separately ; and also so bad17 printed as to be almost unintelligible ; circumstances both of whit have probably contributed to keep it, as it has been kept, totally out of view. Cardan's theory is perfectly false : he supposes, for example, that since there are six faces to a die, it will happen in the long run that each face will come up once in six throws, which is true when many collections of six throws are averaged; but from this he draws the false conclusion that it is an even chance that any one face comes up in three throws. His numerical reasoning is therefore totally incorrect ; but his notions on the general subject of probability are reasonably sound. Fortune, according to him, does not decide the general average of the play, but only the deviations on one side and the other which a small number of cases present ; and experiment would, he prove that the long run would agree with the predictions of theory ; it were to be wished that he had tried it on Ins own theory. This treatise was written about 1564, and published in 1663. But before this, in 1657, the tract of Dc Itatiociniis in Ludo Alex,' was published as an appendix to Schooten's 'Exercitationes Geometrime; being not only the first regular treatise, but the first which applies the theory to chances of less or gain. It was translated into English, with additions, in 1692, the reputed translator being Mottle the secretary of the Royal Society. Then followed the 'Analyse des Jeux de Hasard,' by 31entinort (first edition 1708, second, enlarged, 1713), a work of higher mathematical pretensions. The ' Ars Conjectandi,' of James Bernoulli, pesthu mouely published by his nephew Nicolas, in 1713 (and which, it may be worth noting, is not contained in the collection of James Bernoulli's works), gives the first glimpse of the more difficult class of problems in which processes containing very large numbers are abbreviated by mathematical analysis. This wan carried still further by De Moivre, whose first work, a paper,' Do llensura Sortie' (` Phil. Trans.,' 1711), was expanded into his celebrated treatise on the doctrine of chances, first edition 1718 (not 1716, as frequently stated), second edition 1738, third edition, with his ' Treatise on Life Annuities,' 1756. The next step was made by Bayes (' Phil. Trans.,' 1763 and 1764), who first considered the probability of hypotheses as deduced from observed events.