PROBABILITY, THE MATILEMATTAL THEORY OF. Of all' mathematical theories which can be made in any sense popular, this is perhaps the least generally undertood. There are several reasons for this curious fact, of which we may mention one or two. Pirst.—As by far the simplest and most direct elementary illustrations of its principles are furnished by games of chance, these have been almost invariably used by writers on the subject; and the result has been a popular delusion, to the effect that the theory tends directly to the encouragement of gambling. can be more false than such an idea. Independent of moral considerations, with which we have nothing to do bore, no arguments against gambling can be furnished at all comparable in power with those deduced from the mathematical analysis of the chances of the game. many problems, some of them among tho easiest in the theory, the very highest resources of mathematics are taxed in order to furnish a solution. One reason is very simple. The solutions, however elementary, involving often nothing but the common rules of arith metic, sometimes lead to results depending upon enormous numbers, and very refined analysis is requisite to deduce easily from theso what would otherwise involve calcula- tions, simple enough in character, but of appalling labor. Higher mathematics here perform, in fact, something analogous to skilled labor in ordinary manufactures. The simplest illustration of this is in the use of LOGAIIITEMS (q.v.), which reduce multiplica tion, division, and extraction of roots to mere addition, subtraction, and division respec tively. Powerful as logarithms are, analysis furnishes instruments almost infinitely more powerful. The large numbers which occur in probabilities are usually in the form of products, and we may exemplify the above remarks as follows.
To find the value of the product 1 . 3. 5. 7, no one would think of using anything but common arithmetic; but if he were required to find the value of 1 . 3. 5 77.9 49, he would probably have recourse to logarithms, merely to avoid useless labor of an ele mentary kind. But hi very simple questions in probabilities, it may be requisite to find (approximately) the value of a product such as 1 . 3.5. 7.9 23999—i.e., that of the first 12.000 odd numbers. No one in his senses would dream of attempting this by ordi nary arithmetic, but it is the mere labor, not the inherent difficulty, which prevents him. Few would even attempt, it by means of logarithms; for, even with their aid, the labor would be very great. It is here that the higher analysis steps in, and helps us easily to a sufficiently accurate approximation to the value of this enormous number. Thus, it appears that this objection to the study of the theory of probabilities is not applicable to their principles, which are very elementary, but to the mere mechanical details of the processes of solution of certain problems. Third.—There are other objections, such as
the (so-called) religious one, that "there is no such thing as chance," and that "to cal culate chances is to deny the existence of an all-ruling Providence," etc. ; hut, like many other similar assertions, these are founded on a total ignorance of the nature of the science; and, therefore, although pernicious, may be safely treated with merited contempt. The authors of such objections remind us of the Irishman who attempted to smash lord Bosses great telescope, because "it is irreligious to pry into the mysteries of nature." It appears to us that the best method of explaining the principles of the stibject within our necessarily narrow limits, will be to introduce definitions, etc.. as they may r be called for, in the course of a few elementary illustrations, instead of elaborately pre mising them.
First Case.—The simplest possible illustrations are supplied by the common process 'of " tossing" a coin, with the result of "head" or " tail.' Put H for head, and T for tail. Now, the result of one toss, unless the coin should fall on its edge (which is practically impossible), must be either H or T.
Also, if the coin be not so fashioned as to be more likely to fall on one side than the ether (as, for instance, is the case with loaded dice), these events are equally likely; or, in technical language, equally probable. To determine numerically the likelihood or the probability of either, we must assign sonic numerical value to absolute certainty. This value is usually taken as unity, so that a probability, if short of absolute certainty, is always represented by a proper fraction. Suppose that p (a proper fraction) represents the probability of H, then evidently p is also the probability of T, because the two events are equally likely. But one or other must happen; hence, the sum of the separate probabilities must represent certainty. That is, 1 p p 1, or p = Thus we have assigned a numerical value to the probability of either H or T, by finding what proportion each bears to certainty, and assigning to the latter a simple numerical value.
Suppose, as a contrast, the coin to be an unfair one, such as those sometimes made for swindling purposes, with H on each side. Then we must have in one toss H or II; i.e., H is certain, or its probability is 1. There is no possibility of 1', and therefore its probability is 0. Absolute impossibility is therefore represented by the numerical value r of the probability becoming zero.
Second case.—Suppose a "fair" coin to be tossed twice in succession. The event must be one of the four: