Now the application of the preceding description to our present subject is as follows :—The beginner in exact science has usually no definite notions as to the end which ho is to arrive at; nor do the terms algebra, geometry, mechanics, &c., any associations beyond a vague notion that they are its of a learned system. But it is impos !tibia that the beginner in the subject of this article should be without an explicit and probably • an exaggerated notion of what ho is to attain. There is no unknown Creek or Arabic term the meaning of which must be slowly learned by the study of the science of which it is the nine; the word probability, so well known in the common afihirs of life, stares him in the face at the head of every page, and reminds him to be dissatisfied with the extent of power gained, up to the point at which he has arrived. Unless then he can make up his mind to descend, as a student would do who, having in his head the theory of gravitation and the laws of light, should lay by these grand ideas, and set himself to trace the consequences of the simple notion that two straight lines cannot enclose a space—he must be warned that lie will be likely to quit the subject in diegust. We now proceed to the fundamental points of the theory.
That opinion may be formed with more or less strength, particularly when the subject-matters are of different species, is well known to every one from his own experience. The most decided republican in England, for instance, is not so sure of the wisdom of the Long Par liament as he is that all its members are now dead; and no royalist, however well persuaded of his tenets, thinks the Restoration was of as much consequence to this country as sun, wind, and rain. It matters nothing that the different degrees of assurance refer to very different matters, and are obtained in very different ways ; that they are separate amounts of the same kind of feeling is universally felt and admitted. To make something like a gauge for these degrees of belief is not difficult ; to apply it is a harder task, seeing that the cases which present circumstances of sufficiently definite character are seldom met with.
Suppose a box to contain 3 white and 4 black halls ; it is easily admitted that it is more likely that a black ball should bo drawn than a white one, on the supposition that the drawer does not see the balls. Or rather we should say it is easily admitted that every well regulated mind ought to think a black ball more likely than a white one : and that if any one should imagine the contrary, he has formed an opinion from prejudice, fancy, or want of proper consideration. Just as we should say that if all the balls were black, a black ball would certainly bo drawn, so when a majority of the balls is black, and each one ball is as likely to be drawn as any other, there are more ways of drawing black than white, and we look upon the former as more obtainable, and more likely to be obtained, than the Latter. Common experience makes us consider the black as more likely than white, when the number of black balls is much greater than that of white balls ; as, if there were only 3 white balls, and a million of black ones. Here, as in other questions of magnitude, we can see a difference when the difference is great, which we must perhaps learn to see when it is small : it is plain enough that the black is more likely than the white when there are a million of black balls to one white ; but not so easily grasped that the black is more likely than the white when there are five hundred thousand and one black balls to five hundred thousand white.
The next step to be made is the assertion that when there are 3 white and 4 black balls, the probability of drawing whito is to that of drawing black in the proportion of 3 to 4; that rs, if we could by a voluntary act make our impressions about the probability of future events of that strength which our reason tells us they ought to have, we should choose to expect a black ball more strongly than a white one in the proportion of 4 to 3. The principle on which we do this is the
main point of the theory, the only objectionable part, if there be one : for all the rest is mathematical deduction.
The principle is as follows :—When any number of events, a, n, c, &e., are such that one and only one can happen nt a time, and when a, b, c, &c., are the numbers of ways in which they can severally happen, the probabilities of the several events are in the proportions of the numbers a, b, c, &c. Returning to the preceding simple Instance, wo have an obvious negative reason for supposing that the probabilities should be as 4 to 3, since there is no imaginable ground for assuming, while the excess of black balls is the sole cause of the superior pro bability of drawing one of them, that this excess of probability should be in any other proportion than that of the excess of black balls. If we grant the following, namely, that the probability of having one or other out of two of the different results which a trial may give, is, or ought to be, the suns • of the probabilities of the two separately, we shall be obliged to admit positive reason for the preceding principle, as fellows :—Suppose it box to contain 10 balls, marked 1, 2, &e. tip to 10, and no others. A ball is to be drawn, and the drawer has in his mind an amount of hope, fear, or simple admission of possibility, as the ease may be, as to the happening of each number. If the drawing of No. 1 be to gain him a prize, there is a certain amount of hope ; if it be to procure him a loss, of fear; if neither one nor the other, of feeling that Do. 1 may be that which is drawn. Now let either 1 or 2 bring the gain or loss; is the feeling of hope or fear doubled in strength I or rather, ouviit it to bo doubled! Ho who admits this, admits the whole theory of probabilities, for all the rest is mathematical deduction. Let x be the proper numerical measure of the probability of drawing 1, or of drawing 2, rte. ; these probabilities being equal, since there is, by the hypothesis, nothing to render one more likely than the other. Then, if the preceding be admitted, 2x is the probability that either 1 or 2 is drawn ; 3x that either 1, 5, or 3 is drawn ; and so on up to 10x, which is the probability that one or other of the set 1, 2, 3, 10, shall be drawn. But since that a. drawing shall take place is an abeelnte condition, one of the ten numbers must be drawn ; hence x must be so taken that 10x shall be the numerical measure of certainty. It is indifferent what number is taken to stand for the exponent of certainty, so far as principles are concerned; but in a mathematical point of view, unity is more convenient than anything else. Let unity be adopted, then 10x=1, or x= Hence the chance of draw ing, say one of the three, 1, 2, 3, is --„; and that of drawing one of the 10 remaining 7 is1-. If then the first three should be white balls, and 10 the last seven black, the chance of a white ball is 10 - , that of a black 3 one is -7 ' • and a black ball is more likely than a white one in 10 the proportion of 7 to 3, which being inequality;in the proportion of 21 to 1, the odds are said to be 21 to one in favour of a black ball, or against a white one. By this we do not mean that every man does, in such a case, look for a black ball with an expectation 21 times as great as his expectation of a white ball, but that, if he could measure the strength of his own feelings and adjust them with mathematical precision, he would proportion the strength of the two expectations in the preceding manner. And if money were to be spent upon the expectations, he may as reasonably give 21/. for a black ball, before it appears, as 11. for a white one. We do not naturally reckon probabi lities by numbers ; but nevertheless, we have some kind of estimating apparatus in our brains : Kant called it a teeiOing machine with unstamped weights.