165. But the denominator of this numerator is not the same as
before, but less by half the number of null differences, that is 5.
We thus obtain for the required expectation 165/50 = 3.3.
59. A simple verification of this prediction may thus be obtained. In a table of logarithms note any two digits so situated as to afford no presumption of close correlation; for instance, in the last place of the logarithm of 10009 the digit 7 and in the last lace of the logarithm of 10019 the digit 4, and take the difference between these two, viz. 3, irrespective of sign. Proceed similarly with the similarly situated pair which form the last places of the logarithms of 10029 and 10039; for which the difference is 1, and so on. The mean of the differences thus found ought to be approximately 3.3. Experimenting thus on the last digits of logarithms, in Hutton's tables extending to seven places, from the logarithm of 10009 to the logarithm of 10909, the writer has found for the mean of 250 differences, 3.2.
60. Points taken at Random.—By parity of reasoning it may be shown that if two different milestones are taken at random on a road n miles long (there being a stone at the starting-point) their average distance apart is ⅓(n + 2).
61. If instead of finite differences as in the last two problems the intervals between the numbers or degrees which may be selected are indefinitely small, we have the theorem that the mean distance between two points taken at random on a finite straight line is a third of the length of that straight line.
62. The fortuitous division of a straight line is happily employed by Professor Morgan Crofton to exhibit Laplace's method of Rules for Voting at Elections. determining the worth of several candidates by combining the votes of electors. There is a close relation between this method and the method above given for determining the probabilities of several alternatives by combining the judgments of different judges.[1] But there is this difference—that the several estimates of worth, unlike those of probability, are not subject to the condition that their sum should be equal to a constant quantity (unity). The quaesita are now expectations, not probabilities. Professor Morgan Crofton's version.[2] of the argument is as follows. Suppose there are n candidates for an office; each elector is to arrange them in what he believes to be the order of merit; and we have first to find the numerical value of the merit he thus implicitly attributes to each candidate. Fixing on some limit a as the maximum of merit, n arbitrary values less than a are taken and then arranged in order of magnitude—least, second, third, . . . greatest; to find the mean value of each.
|
| ||||||||
| A | X | Y | Z | B | ||||
Take a line AB = a, and set off n arbitrary lengths AX, AY, AZ . . . beginning at A; that is, n points are taken at random in AB. Now the mean values of AZ, XY, YZ, . . . are all equal; for if a new point P be taken at random, it is equally likely to be 1st, 2nd, 3rd, &c., in order beginning from A, because out of n + 1 points the chance of an assigned one being 1st is (n + 1)−1; of its ing 2nd (n + 1)−1; and so on. But the chance of P being 1st is equal to the mean value of AX divided by AB; of its being 2nd M(XY) ÷ AB; and so on. Hence the mean value of AX is AB (n + 1)−1; that of AY is 2AB (n + 1)−1; and so on. Thus the mean merit assigned to the several candidates is
a(n + 1)−1, 2a(n + 1)−1, 3a(n + 1)−1 . . . na(n + 1)−1.
Thus the relative merits may be estimated by writing under the names of the candidates the numbers 1, 2, 3, . . . n. The same being done by each elector, the probability will be in favour of the candidate who has the greatest sum.
Practically it is to be feared that this plan would not succeed, because, as Laplace observes, not only are electors swayed by many considerations independent of the merit of the candidates, but they would often place low down in their list any candidate whom they judged a formidable competitor to the one they preferred, thus giving an unfair advantage to candidates of mediocre merit.
63. This objection is less appropriate to competitive examinations, to which the method may seem applicable. But there is a more fundamental objection in this case, if not indeed in every case, to the reasoning on which the method rests: viz. that there is supposed an a priori distribution of values which is in general not supposable; viz. that the several estimates of worth, the marks given to different candidates by the same examiner, are likely to cover evenly the whole of the tract between the minimum and maximum, e.g. between 0 and 100. Experience, fortified by theory, shows that very generally such estimates are not thus indifferently disposed, but rather in an order which will presently be described as the normal law of error.[3] The theorem governing the case would therefore seem to be not that which is applied by Laplace and Morgan Crofton, but that which has been investigated by Karl Pearson,[4] a theorem which does not lend itself so readily to the purpose in hand.[5]
64. Expectation of Advantage.—The general examples of expectation which have been given may be supplemented by some appropriate to that special use of the term which Laplace has sanctioned when he considers the subject of expectation as a “good”; in particular money, or that for the sake of which money is desired, “moral” advantage, in more modern phrase utility or satisfaction.
65. Pecuniary Advantage.—The most important calculations of pecuniary expectation relate to annuities and insurance; based largely on life tables from which the expectation of life itself, as well as of money value at the end, or at any period, of life is predicted. The reader is referred to these heads for practical exemplifications of the calculus. It must suffice here to point out how the calculations are facilitated by the adoption of a law of frequency, the Gompertz or the Gompertz-Makeham law, which on the one hand can hardly be ranked with hypotheses resting on a vera causa, yet on the other hand is not purely empirical, but is recommended, as germane to the subject-matter, by colourable suppositions.[6]
66. There is space here only for one or two simple examples of money as the subject of expectation. Two persons A and B throw a die alternately, A beginning, with the understanding that the one who first throws an ace is to receive a prize of £1. What are their respective expectations?[7] The chance that the prize should be won at the first throw is 16, the chance that it should be won at the second throw is 56 16; at the third throw (56)2 16, at the fourth throw (56)3 16, and so on. Accordingly the expectation of A
= £1 × 16 {1 + (56)2 + (56)4 + . . .};
of B
= £1 × 16 ⋅ 56 {1 + (56)2 + (56)4 + . . .};
Thus A's expectation is to B's as 1 : 56. But their expectations must together amount to £1. Therefore A's expectation is 611 of a pound, B's 511.
67. There are n tickets in a bag, numbered 1, 2, 3, . . . n. A man draws two tickets at once, and is to receive a number of sovereigns equal to the product of the numbers drawn. What is his expectation?[8] It is the number of pounds divided by an improper fraction of which the denominator is the number of possible products, ½n(n − 1), and the numerator is the sum of all possible products = ½{(1 + 2 + 3 . . . + n)2 − (12 + 22 + . . . + n2)}. Whence the required number (of pounds) is found to be 112(n + 1)(3n + 2). The result may be contrasted with what it would be the two tickets were not to be drawn at once, but the second after replacement of the first. On this supposition the expectation in respect of one of the tickets separately is ½(n + 1). Therefore, as the two events are now independent, the expectation of the product,[9] being the product of the expectations, is {½(n + 1)}2.
68. Peter throws three coins, Paul two. The one who obtains the greater number of heads wins £1. If the number of heads are equal, they play again, and so on, until one or other obtains a greater number of heads. What are their respective expectations?[10] At the first trial there are three alternatives: (α) Peter obtains more heads than Paul, (β) an equal number, (γ) fewer. The cases in favour of α are (1) Peter obtains three heads, (2) Peter, two heads, while Paul one or none, (3) Peter one head, Paul none. The cases in favour of β are (1) two heads for both, or (2) one head, or (3) none, for both. The remaining case favours γ. The probability of α is 18 + 38 34 + 38 14 = 12. The probability of β is 38 14 + 38 12 + 18 14 = 516. The probability of γ is 1 − 1316 = 316. Alternative, β is to be split up into three α′, β′, γ′, of which the probabilities (when β has occurred) are as before, 816, 516, 316. β′ is similarly split up, and so on. Thus Peter's expectation is 816{1 + 516 + (516)2 + . . .}£1 = 811£1. Paul's expectation is 811£1.
An urn contains m balls marked 1, 2, 3, . . . m. Paul extracts successively the m balls, under an agreement to give Peter a shilling every time that a ball comes out in its proper order. What is Peter's expectation? The expectation with respect to any one ball is 1m, and therefore the expectation with respect to all is 1 (shilling).[11]
69. Advantage subjectively estimated.—Elaborate calculations are paradoxically employed by Laplace and other mathematicians to determine the expectation of subjective advantage in various cases of risk. The calculation is based on Daniel Bernoulli's formula which may be written thus: If x denote a man's physical fortune, and y the corresponding moral fortune
y = k log (x/h),
k, h being constants. x and y are always positive, and x > h; for every
- ↑ Above par. 52.
- ↑ Loc. cit. § 45.
- ↑ See Edgeworth, “Elements of Chance in Examinations,” Journ. Stat. Soc. (1890). Cf. below, par. 124.
- ↑ Biometrika, i. 390.
- ↑ Moore, of Columbia University, New York, has attempted to trace Karl Pearson's theory in the statistics relating to the efficiency of wages (Economic Journal, Dec. 1907; and Journ. Stat. Soc., Dec. 1907).
- ↑ Cf. below, par. 169.
- ↑ Whitworth, Choice and Chance, question 126.
- ↑ Whitworth, Exercises, No. 567.
- ↑ According to the principle above enounced, par. 15.
- ↑ Bertrand, id. § 44, prob. xlvii.
- ↑ Bertrand, id. § 39, prob. xliii. It is not to be objected that the probabilities on which the several expectations are calculated are not independent (above, par. 16).
