Page:EB1911 - Volume 22.djvu/391

“The probability of an event is the ratio of the number of cases which favour it to the number of all the possible cases, when nothing leads us to believe that one of these cases ought to occur rather than the others; which renders them, for us, equally possible.”^[1] Against this view it is urged that merely psychological facts can at best afford a measure of belief, not of credibility. Accordingly, the ground of probability is sought in the observed fact of a class or “series”^[2] such that if we take a great many members of the class, or terms of the series, the members thereof which belong to a certain assigned species compared with the total number taken tends to a certain fraction as a limit. Thus the series which consists of heads and tails obtained by tossing up a well-made coin is such that out of a large number of throws the proportion giving heads is nearly half.

3. These views are not so diametrically opposed as may at first appear. On the one hand, those who follow Laplace would of course admit that the presumption afforded by the “number of favourable cases” with respect to the probability of throwing either five or six with a die must be modified in accordance with actual experience such as that below cited^[3] respecting particular dice that they turn up five or six rather oftener than once in three times. On the other hand, the series which is regarded as the empirical basis of probability is not a simple matter of fact. There are implied conditions which are not satisfied by the sort of uniformity which ordinarily characterizes scientific laws; which would not be satisfied for instance by the proportionate frequency of any one digit, e.g. 8, in the expansion of any vulgar fraction, though the expression may consist of a circulating decimal with a very long period.^[4]

4. The type of the series is rather the frequency of the several digits in the expansion of an incommensurable constant such as √2, log 11, π, &c.^[5] The observed fact that the digits occur with equal frequency is fortified by the absence of a reason why one digit should occur oftener than another.^[6]

5. The most perfect types of probability appear to present the two aspects: proportion of favourable cases given a priori and frequency of occurrence observed a posteriori. When one of these attributes is not manifested it is often legitimate to infer its existence from the presence of the other. Given numerous batches of balls, each batch numbering say 100 and consisting partly of white and partly of black balls; if the percentages of white balls presented by the set of batches averaged, and, as it were, hovered about some particular percentage, e.g. 50, though we knew as an independent datum, or by inspection of the given percentages, that the series was not obtained by simply extracting a hundred balls from a jar containing a mélange of white and black balls, we might still be justified in concluding that the observed phenomenon resulted from a system equivalent to a number of jars of various constitution, compounded in some complicated fashion. So Laplace may be justified in postulating behind frequencies embodied in vital statistics the existence of a “constitution” analogous to games of chance, “possibilities” or favourable cases which might conceivably be “developed” or discussed.^[7] On the other hand, it is often legitimate to infer from the known proportion of favourable cases a corresponding frequency of occurrence. The cogency of the inference will vary according to the degree of experience. That one face of a die or a coin will turn up nearly as often as another might be affirmed with perfect confidence of the particular dice which Weldon threw some thousands of times,^[8] or the coins with which Professor Pearson similarly operated.^[9] It may be affirmed with much confidence of ordinary coins and dice without specific experience, and generally, where fairplay is presumed, of games of chance. This confidence is based not only on experiments like those tried by Buffon, Jevons and many others,^[10] but also on a continuous, extensive, almost unconsciously registered experience in pari materia. It is this sort of experience which justifies our expectation that commonly in mathematical tables one digit will occur as often as another, that in a shower about as many drops will fall on one element of area as upon a neighbouring spot of equal size. Doubtless the presumption must be extended with caution to phenomena with which we are less familiar. For example, is a meteor equally likely to hit one square mile as another of the earth's surface? We seem to descend in the scale of credibility from absolute certainty that alternative events occur with about equal frequency to absolute ignorance whether one occurs more frequently than the other. The empirical basis of probability may appear to become evanescent in a case like the following, which has been discussed by many writers on Probabilities.^[11] What is the probability of drawing a white ball from a box of which we only know that it contains balls both black and white and none of any other colour? In this case, unlike the case of an urn containing a mixture of white and black balls in equal proportions, we have no reason to expect that if we go on drawing balls from the urn, replacing each ball after it has been drawn, that the series so presented will consist of black and white in about equal numbers. But there is ground for believing that in the long course of experiences in pari materia—other urns of similar constitution, other cases in which there is no reason to expect one alternative more than another—an event of one kind will occur about as often as one of another kind. A “cross-series”^[12] is thus formed which seems to rest on as extensive if not so definite an empirical basis as the series which we began by considering. Thus the so-called “intellectual probability”^[13] which it has been sought to separate from the material probability verified by frequency of occurrence, may still rest on a similar though less obvious ground of experience. This type of probability not verified by specific experience is presented in two particularly important classes.

6. Unverified Probabilities.—In applying the theory of errors to the art of measurement it is usual to assume that prior to observation one value of the quantity under measurement is as likely as another. “When the probability is unknown,” says Laplace,^[14] “we may equally suppose it to have any value between zero and unit.” The assumption is fundamentally similar whether the quantum is a ratio to be determined by the theorem of Bayes,^[15] or an absolute quantity to be determined by the more general theory of error. Of this first principle it is well observed by Professor Karl Pearson^[16]: “There is an element of human experience at the bottom of Laplace's assumption.” Professor Pearson quotes with approbation^[17] the following account of the matter: “The assumption that any probability-constant about which we know nothing in particular is as likely to have one value as another is grounded upon the rough but solid experience that such constants do as a matter of fact as often have one value as another.”

7. It may be objected, no doubt, that one value (of the object under measurement) is often known beforehand not to be as likely as another. The barometric height for instance is not equally likely to be 29 in. or to be 2 in. The reply is that the postulate is only required with respect to a small tract in a certain neighbourhood, some 2 in. above and below 291/2 in. in the case of barometric pressure.

8. It is further objected that the assumption in question involves inconsistencies in cases like the following. Suppose observations are made on the length of a pendulum together with the time of its oscillation. As the time is proportional to the square root of the length, it follows that if the values of the length occur with equal frequency those of the time cannot do so; and, inversely, if the proposition is true of the times it cannot be true of the lengths.^[18] One reply to this objection is afforded by the reply to the former one. For where we are concerned only with a small tract of values it will often happen that both the square and the square root and any ordinary function of a quantity which assumes equivalent values with equal probability will each present an approximately equal distribution of probabilities.^[19] It may further be replied that in general the reasoning does not require the a priori probabilities of the different values to be very nearly equal; it suffices that they should not be very unequal;^[20] and this much seems to be given by experience.

9. Whenever we can justify Laplace's first principle^[21] that “probability is the ratio of the number of favourable cases to the number of all possible cases” no additional difficulty is involved in his second

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1. Laplace, Théorie analytique des probabilités, liv. II. ch. i. No. 1. Cf. Introduction, IIᵉ principe.
2. The term employed by Venn in his important Logic of Chance.
3. Below, par. 119.
4. E.g. 1/1861 in the expansion of which the digit 8 occurs once in ten times in seemingly random fashion (see Mess. of Maths. 1864, vol. 2, pp. 1 and 39).
5. The type shows that the phenomena which are the object of probabilities do not constitute a distinct class of things. Occurrences which perfectly conform to laws of nature and are capable of exact prediction yet in certain aspects present the appearance of chance. Cf. Edgeworth, “Law of Error,” Cam. Phil. Trans., 1905, p. 128.
6. Cf. Venn. op. cit. ch. v. § 14; and v. Kries on the “Prinzip des mangelnden Grundes” in his Wahrscheinlichkeitsrechnung, ch. i. § 4, et passim.
7. In a passage criticized unfavourably by Dr Venn, Logic of Chance, ch. iv. § 14.
8. Below, par. 115.
9. Chances of Death, i. 44.
10. A summary of such experiments, comprising above 100,000 trials, is given by Professor Karl Pearson in his Chances of Death, i. 48.
11. E.g. J. S. Mill, Logic, bk. III., ch. xviii. § 2.
12. Cf. Venn, Logic of Chance, ch. vi. § 24.
13. Boole, Trans. Roy. Soc. (1862), ix. 251.
14. Op. cit. Introduction.
15. Below, par. 130.
16. Grammar of Science, ed. 2, p. 146.
17. From the article by the present writer on the “Philosophy of Chance” in Mind, No. ix., in which some of the views here indicated are stated at greater length than is here possible.
18. Cf. v. Kries, op. cit. ch. ii.
19. On the principle of Taylor's theorem; cf. Edgeworth, Phil. Mag. (1892), xxxiv. 431 seq.
20. Cf. J. S. Mill, in the passage referred to below, par. 13, on the use that may be made of an “antecedent probability,” though “it would be impossible to estimate that probability with anything like numerical precision.”
21. Op. cit. Introduction.