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}, 2*a*(*n* + 1)^{−1}, 3*a*(*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, ½*n*(*n* − 1), and the numerator is the sum of all
possible products = ½{(1 + 2 + 3 . . . + *n*)^{2} − (1^{2} + 2^{2} + . . . + *n*^{2})}.
Whence the required number (of pounds) is found to be 112(*n* + 1)(3*n* + 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 1*m*, 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).