precise, yet something worth knowing. The a priori improbability
of the maker's inaccuracy must be very great to overcome the
improbability of such an event occurring by chance if the machine is
accurately made (accuracy being defined, say, by the condition
that the ratio of red to [red + white] would prove to be in the indefinitely
long run of trials between 0.499 and 0.501). The odds
against the so defined event occurring are found to be some millions
to one.^{[1]}
47. The difficulty recurs in more practical problems: for instance, certain symptoms having been observed, to find the probability that they are produced by a particular disease. Such concrete applications of probabilities are often open to the sort of objections which have been urged against the classical use of the calculus to determine the probability that witnesses are true, or judges just.
48. Probability of Testimony.—The application of probabilities to testimony proceeds upon two assumptions: (1) that to each witness there pertains a coefficient of probability representing the average frequency with which he speaks the truth or untruth, (2) that the statements of witnesses are independent in the sense proper to probabilities. Thus if two witnesses concur in making a statement which must be either true or false, their agreement is a circumstance which is only to be accounted for by one of two alternatives: either that they are both speaking the truth, or both false. If the average truthfulness—the credibility—of one witness is p, that of the other p′, then the probabilities of the two alternative explanations are to each other in the ratio pp′ : (1 − p)(1 − p′); the probability that the statement is true is pp′/{pp′ + (1 − p)(1 − p′)}. So far no account is taken of the a priori probability of the statement. This evidence may be treated as an independent witness. Thus, if a person whose credibility is p asserts that he has seen at whist a hand consisting entirely of trumps dealt from a wellshuffled pack of cards, there are two alternative explanations of his assertion, with probabilities in the ratio
p × 0.000,000,000,0063 : (1  p) × 0.999,999,999,993.
The truthfulness of the witness must be very great to outweigh the a priori improbability of the fact.^{[2]} These formulae are easily extended to the case of three or more witnesses. The probability of a statement made by three witnesses of respective credibilities p, p′, p″ is
pp′p″/{pp′p″ + (1 − p)(1 − p′)(1 − p″)}.
For r witnesses we have
p_{1}p_{2} ⋅ ⋅ ⋅ p_{r} / {p_{1}p_{2} ⋅ ⋅ ⋅ p_{r} + (1 − p_{1})(1 − p_{2}) ⋅ ⋅ ⋅ (1 − p_{r})}.
Dividing both the numerator and the denominator by p_{1}p_{2} ⋅ ⋅ ⋅ p_{r}, we see that the probability of the statement increases with the number of the witnesses, provided that for every witness (1 − p)/p is a proper fraction, and accordingly p > ½. As an example of several witnesses, let us inquire how many witnesses to a fact such as a hand at whist consisting entirely of trumps would be required in order to make it an even chance that the fact occurred, supposing the credibility of each witness to be 910.^{[3]} Let x be the required number of witnesses. We have the 1/(1 + (19)^{x}0.000,000,000,006) = ½, or x log 9 = 12.2. Whence, if x is 13, it is more than an even chance that the statement is true.
49. When an event may occur in two or more ways equally probable a priori, the formulae show that the probability of the statement will depend on the credibility of the witnesses; and accordingly the explicit consideration of a priori probabilities may, as in our first instance, be omitted. One who reports the number of a ticket obtained at a lottery ordinarily makes a statement against which there is no a priori improbability; but if the number is one which had been predicted, there is an a priori improbability 1n that an assigned ticket should be drawn out of a mélange of n tickets. Similar reasoning is applicable to the probability that the decisions of judgments, the verdict of juries, is right.
50. The assumptions upon which all this reasoning is based are open to serious criticisms. The postulated independence of witnesses and judges is frequently not realized. The revolutionary tribunal which condemned Condorcet was affected by an identity of illusions and passions which that mathematician had not taken into account when he calculated “that the probability of a decision being conformable to truth will increase indefinitely as the number of voters is increased.”^{[4]}
51. The use of coefficients based on the average truthfulness or justice of each witness and judge involves the neglect of particulars which ought to influence our estimate of probability, such as the consistency of a witness's statements and the relation of the case to the interests, prejudices and capacities of the witness or the judge.^{[5]} Thus even in so simple a case as the alleged occurrence of an extraordinary hand at whist, the “truthfulness” of the witness in the general sense of the term may not adequately represent his liability to have made a mistake about the shuffling.^{[6]} A neglect of particulars, however, is sometimes practised with success in the applications of statistics (insurance, for instance). Perhaps there are broad results and general rules to which the mathematical theory may be applicable. Perhaps the laborious researches of Poisson on the “probability of judgments" are not, as they have been called by an eminent mathematician, absolument rien.^{[7]} More than mathematical interest may attach to Laplace's investigation of a rule appropriate to cases like the following. An event (suppose the death of a certain person) must have proceeded from one of n causes A, B, C, &c., and a tribunal has to pronounce on which is the most probable. Professor Morgan Crofton's original proof of Laplace's rule is here reproduced.^{[8]}
52. Let each member of the tribunal arrange the causes in the order of their probability according to his judgment, after weighing the evidence. To compare the presumption thus afforded by any one judge in favour of a specified cause with that afforded by the other judges, we must assign a value to the probability of the cause derived solely from its being, say, the rth on his list. As he is supposed to be unable to pronounce any closer to the truth than to say (suppose) H is more likely than D, D more likely than L, &c., the probability of any cause will be the average value of all those which that probability can have, given simply that it always occupies the same place on the list of the probabilities arranged in order of magnitude. As the sum of the n probabilities is always 1, the question reduces to this:—
Any whole (such as the number 1) is divided at random into n parts, and the parts are arranged in the order of their magnitude—least, second, third, . . . greatest; this is repeated for the same whole a great number of times; required the mean value of the least, of the second, &c., parts, up to that of the greatest.
A  B  b  

Let the whole in question be represented by a line AB = a, and let it be divided at random into n parts by taking n − 1 points indiscriminately on it. Let the required mean values be
λ_{1}a, λ_{2}a, λ_{3}a . . . . λ_{n}a,
where λ_{1}, λ_{2}, λ_{3} . . . must be constant fractions. As a great number of positions is taken in AB for each of the n points, we may take a as representing that number; and the whole number N of cases will be
N = a^{n−1}.
The sum of the least parts, in every case, will be
S_{1} = Nλ_{1}a = λ_{1}a^{n}.
Let a small increment, Bb = δa, be added on to the line AB at the end B; the increase in this sum is δS_{1} = nλ_{1}a^{n−1}δa. But, in dividing the new line Ab, either the n − 1 points all fall on AB as before, or n − 2 fall on AB and 1 on Bb (the cases where 2 or more fall on Bb are so few we may neglect them). If all fall on AB, the least part is always the same as before except when it is the last, at the end B of the line, and then it is greater than before by δa; as it falls last in n^{−1} of the whole number of trials, the increase in S_{1} is n^{−1}a^{n−1}δa. But if one point of division falls on Bb, the number of new cases introduced is (n − 1)a^{n − 2}δa; but, the least part being now an infinitesimal, the sum S_{1} is not affected; we have therefore
δS_{1} = nλ_{1}a^{n−1}δa = n^{−1}a^{n−1}δa;
∴λ_{1} = n^{−2}.
To find λ_{2}, reasoning exactly in the same way, we find that where one point falls on Bb and n − 2 on AB, as the least part is infinitesimal, the second least part is the least of the n − 1 parts made by the n − 2 points; consequently, if we put λ_{1}′ for the value of λ_{1} when there are n − 1 parts only, instead of n,
δS_{2} = nλ_{2}a^{n−1}δa = n^{−1}a^{n−1}δa + (n − 1)a^{n−2}λ_{1}′aδa,
∴nλ_{2} = n^{−1} + (n − 1)λ_{1}′; but λ′_{1} = (n − 1)^{−2};
∴nλ_{2} = n^{−1} + (n − 1)^{−1}.
In the same way we can show generally that
nλ_{r} = n^{−1} + (n − 1)λ′_{r−1};
and thus the required mean value of the rth part is
λ_{r}a = an^{−1}{n^{−1} + (n − 1)^{−1} + (n − 2)^{−1} + ⋅ ⋅ ⋅ (n − r + 1)^{−1}}.
 ↑ By a calculation based on the fundamental theorem (above, par. 23; cf. below, par. 103).
 ↑ But see below, par. 51.
 ↑ Morgan Crofton, loc. cit. p. 778, par. 1.
 ↑ Essai, p. 6 (there is postulated a proviso analogous to that which has been stated in par. 49 above, with reference to witnesses: that the probability of any one voter being right is > ½).
 ↑ See Mill's forcible remarks on this use of probabilities, which he places among the “misapplications of the calculus which have made it the real opprobrium of mathematics” (Logic, Book III, ch. xviii. § 3). Cf. Bertrand, Calcul des probabilités; Venn, Logic of Chance, ch. xvi. § 57; v. Kries, Principien der Wahrscheinlichkeitsrechnung, ch. ix., preface, § v., and ch. xiii. §§ 12, 13; Laplace's general reflections on this matter seem more valuable than his calculations: “Tant de passions et d'interêts particuliers y mêlent si souvent leur influence qu'il est impossible de soumettre au calcul cette probabilité,” op. cit. Introduction (Des Choix et décisions des assemblées).
 ↑ As to the possibility of mistake in this respect, see Proctor, How to play Whist, p. 121.
 ↑ Bertrand, loc. cit.
 ↑ Loc. cit. § 43.