we may add together these separate probabilities, and we find that 32 is the complete probability that a white ball will be next drawn under the conditions and data supposed. General Solution of the Inverse Problem. In the instance of the inverse method described in the last section, the balls supposed to be in the ballot-box were few, for the purpose of simplifying the calculation. In order that our solution may apply to natural phenomena, we must render our hypotheses as little arbitrary as possible. Having no à priori knowledge of the conditions of the phenomena in question, there is no limit to the variety of hypotheses which might be suggested. Mathematicians have therefore had recourse to the most extensive suppositions which can be made, namely, that the ballot-box contains an infinite number of balls; they have then varied the proportion of white to black balls continuously, from the smallest to the greatest possible proportion, and estimated the aggregate probability which results from this comprehensive supposition. To explain their procedure, let us imagine that, instead of an infinite number, the ballot-box contains a large finite number of balls, say 1000. Then the number of white balls might be 1 or 2 or 3 or 4, and so on, up to 999. Supposing that three white and one black ball have been drawn from the urn as before, there is a certain very small probability that this would have occurred in the case of a box containing one white and 999 black balls; there is also a small probability that from such a box the next ball would be white. Compound these probabilities, and we have the probability that the next ball really will be white, in consequence of the existence of that proportion of balls. If there be two white and 998 black balls in the box, the probability is greater and will increase until the balls are supposed to be in the proportion of those drawn. Now 999 different hypotheses are possible, and the calculation is to be made for each of these, and their aggregate taken as the final result. It is apparent that as the number of balls in the box is increased, the absolute probability of any one hypothesis concerning the exact proportion of balls is decreased, but the aggregate results of all the hypotheses will assume the character of a wider average. When we take the step of supposing the balls within the urn to be infinite in number, the possible proportions of white and black balls also become infinite, and the probability of any one proportion actually existing is infinitely small. Hence the final result that the next ball drawn will be white is really the sum of an infinite number of infinitely small quantities. It might seem impossible to calculate out a problem having an infinite number of hypotheses, but the wonderful resources of the integral calculus enable this to be done with far greater facility than if we supposed any large finite number of balls, and then actually computed the results. I will not attempt to describe the processes by which Laplace finally accomplished the complete solution of the problem. They are to be found described in several English works, especially De Morgan's Treatise on Probabilities, in the Encyclopædia Metropolitana, and Mr. Todhunter's History of the Theory of Probability. The abbreviating power of mathematical analysis was never more strikingly shown. But I may add that though the integral calculus is employed as a means of summing infinitely numerous results, we in no way abandon the principles of com binations already treated. We calculate the values of infinitely numerous factorials, not, however, obtaining their actual products, which would lead to an infinite number of figures, but obtaining the final answer to the problem by devices which can only be comprehended after study of the integral calculus. It must be allowed that the hypothesis adopted by Laplace is in some degree arbitrary, so that there was some opening for the doubt which Boole has cast upon it.1 But it may be replied, (1) that the supposition of an infinite number of balls treated in the manner of Laplace is less arbitrary and more comprehensive than any other that can be suggested. (2) The result does not differ 1 Laws of Thought, pp. 368-375. much from that which would be obtained on the hypothesis of any large finite number of balls. (3) The supposition leads to a series of simple formulas which can be applied with ease in many cases, and which bear all the appearance of truth so far as it can be independently judged by a sound and practiced understanding. Rules of the Inverse Method. By the solution of the problem, as described in the last section, we obtain the following series of simple rules. 1. To find the probability that an event which has not hitherto been observed to fail will happen once more, divide the number of times the event has been observed increased by one, by the same number increased by two. If there have been m occasions on which a certain event might have been observed to happen, and it has happened on all those occasions, then the probability that it will m + happen on the next occasion of the same kind is m + 2 For instance, we may say that there are nine places in the planetary system where planets might exist obeying Bode's law of distance, and in every place there is a planet obeying the law more or less exactly, although no reason is known for the coincidence. Hence the probability that the next planet beyond Neptune will conform to the law is 1. 2. To find the probability that an event which has not hitherto failed will not fail for a certain number of new occasions, divide the number of times the event has happened increased by one, by the same number increased by one and the number of times it is to happen. An event having happened m times without fail, the probability that it will happen n more times is m+ I m + n + 1 Thus the probability that three new planets would obey Bode's law is 19; but it must be allowed that this, as well as the previous result, would be much weakened by the fact that Neptune can barely be said to obey the law. 3. An event having happened and failed a certain number of times, to find the probability that it will happen the next time, divide the number of times the event has S happened increased by one, by the whole number of times the event has happened or failed increased by two. If an event has happened m times and failed n times, the probability that it will happen on the next occasion is m+I m+n+2' Thus, if we assume that of the elements discovered up to the year 1873, 50 are metallic and 14 nonmetallic, then the probability that the next element discovered will be metallic is. Again, since of 37 metals which have been sufficiently examined only four, namely, sodium, potassium, lanthanum, and lithium, are of less density than water, the probability that the next metal examined or discovered will be less dense than water is 4 + I 37 +2 or 5 39 We may state the results of the method in a more general manner thus,1-If under given circumstances certain events A, B, C, &c., have happened respectively m, n, p, &c., times, and one or other of these events must happen, then the probabilities of these events are proportional to m + 1, n + 1, p + 1, &c., so that the probability of A will be But if new m+ I m + 1 + n + I + p + 1 + &c. events may happen in addition to those which have been observed, we must assign unity for the probability of such new event. The odds then become I for a new event, m+1 for A, n + 1 for B, and so on, and the absolute m + I probability of A is I + m + I + n + 1 + &c. It is interesting to trace out the variations of probability according to these rules. The first time a casual event happens it is 2 to 1 that it will happen again; if it does happen it is 3 to 1 that it will happen a third time; and on successive occasions of the like kind the odds become 4, 5, 6, &c., to I. The odds of course will be discriminated from the probabilities which are successively 3,,, &c. Thus on the first occasion on which a person sees a shark, and notices that it is accompanied by a little pilot fish, the odds are 2 to 1, or the probability, that the next shark will be so accompanied. 1 De Morgan's Essay on Probabilities, Cabinet Cyclopædia, p. 67. When an event has happened a very great number of times, its happening once again approaches nearly to certainty. If we suppose the sun to have risen one thousand million times, the probability that it will rise again, on 1,000,000,000 + I the ground of this knowledge merely, is 1,000,000,000+ 1+1 But then the probability that it will continue to rise for as 1,000,000,000 + I long a period in the future is only or almost 2,000,000,000 + I' exactly. The probability that it will continue so rising a thousand times as long is only about 10. The lesson which we may draw from these figures is quite that which we should adopt on other grounds, namely, that experience never affords certain knowledge, and that it is exceedingly improbable that events will always happen as we observe them. Inferences pushed far beyond their data soon lose any considerable probability. De Morgan has said,1" No finite experience whatsoever can justify us in saying that the future shall coincide with the past in all time to come, or that there is any probability for such a conclusion." On the other hand, we gain the assurance that experience sufficiently extended and prolonged will give us the knowledge of future events with an unlimited degree of probability, provided indeed that those events are not subject to arbitrary interference. It must be clearly understood that these probabilities are only such as arise from the mere happening of the events, irrespective of any knowledge derived from other sources. concerning those events or the general laws of nature. All our knowledge of nature is indeed founded in like manner upon observation, and is therefore only probable. The law of gravitation itself is only probably true. But when a number of different facts, observed under the most diverse circumstances, are found to be harmonized under a supposed law of nature, the probability of the law approximates closely to certainty. Each science rests upon so many observed facts, and derives so much support from analogies or connections with other sciences, that there are comparatively few cases where our judgment of the probability of an event depends entirely upon a few ante 1 Essay on Probabilities, p. 128. |