of eye or curliness of hair. A closer analogy is supplied by the older writers who boldly handle “moral” or subjective advantage, as will be shown under the next head.
15. (3) Axioms of Expectation.—Expectation so far as it involves probability presents the same philosophical questions. They occur chiefly in connexion with two principles analogous to and deducible from propositions which have been stated with respect to probability. (i.) The expectation of the sum of two quantities subject to risk is the sum of the expectations of each. (ii.) The expectation of the product of two quantities subject to risk is the product of the expectations of each; provided that the risks are independent. For example, let one of the fortuitously fluctuating quantities be the winnings of a player at a game in which he takes the amount A if he throws ace with a die (and nothing if he throws another face). Then the expectation of that quantity is 1A; or, in n trials (n being large), the player may expect to win about n1A. Let the other fortuitously fluctuating quantity be winnings of a player at a game in which he takes the amount B when an ace of any suit is dealt from an ordinary pack of cards. The expectation of this quantity is 1B; or in n trials the player may expect to win about n1B. Now suppose a compound trial at which one simultaneously throws a die and deals a card; and let his winning at a compound trial be the sum of the amounts which he would have received for the die and the card respectively at a simple trial. In n such compound trials he may expect to win about n1A + n1B, or the expectation of the winning at a compound trial is the sum of the separate expectations. Next suppose the winning at a compound trial to be the product of the two amounts which he would have received for the die and the card if played at a simple trial. It is zero unless the player obtains two aces. It is A × B when this double event occurs. But this double event occurs in the long run only once in 78 times. Accordingly the expectation of the winning at a compound trial at which the winning is the product of the winnings at two simple trials is the product of the separate expectations. What has been shown for two expectations of the simplest type, where α is the probability of an event which has been associated with a quantity a, may easily be extended to several expectations each of the type
a1α1 + a2α2 + a3α3 + . . .
where arαr is an expectation of the simplest type, above exemplified, or of the type a1α1 × a2α2 × a3α3 × . . . or a mixture of these types. For by the law which has been exemplified the sum of r expectations can always be reduced to the sum of r − 1, and then the r − 1 to r − 2, and so on; and the like is true of products.
16. It should be remarked that the proviso as to the independence of the probabilities involved is required only by the second of the two fundamental propositions. It may be dispensed with by the first. Thus in the example of interdependent probabilities given by Laplace—three urns about which it is known that two contain only black balls and one only white—if a person drawing a ball first from C and then from B is to receive x shillings every time he draws a white ball, from one or other of the urns, he may expect if he performs the compound operation n times to receive n × 2 × ⅔x shillings. But the expectation of the product of the number of shillings won by drawing a white ball from C and the number of shillings won by afterwards drawing a white ball from B is not n(⅔)2x2, but n⅓x2.
17. The first of the two principles is largely employed in the practical applications of probabilities. The second principle is largely employed in the higher generalizations of the science (the laws of error demonstrated in Part II.); the requisite independence of the involved probabilities being mostly of the unverified species.
18. Expectation of Utility.—A philosophical difficulty peculiar to expectation arises when the quantity expected has not the objective character usually presupposed in the applications of mathematics. The most signal instance occurs when the expectation relates to an advantage, and that advantage is estimated subjectively by the amount of utility or satisfaction afforded to the possessor. Mathematicians have commonly adopted the assumption made by Daniel Bernoulli that a small increase in a person's material means or “physical fortune” causes an increase of satisfaction or “moral fortune,” inversely proportional to the physical fortune; and accordingly that the moral fortune is equateable to the logarithm of the physical fortune. The spirit in which this assumption should be employed is well expressed by Laplace when he says that the expectation of subjective advantage (l'espérance morale) “depends on a thousand variable circumstances which it is almost always impossible to define and still more to submit to calculation.” “One cannot give a general rule for appreciating this relative value,” yet the principle above stated in “applying to the commonest cases leads to results which are often useful.”
19. In this spirit we may regard the logarithm in Bernoulli's (as in Malthus's) theory as representative of a more general relation. Thus generalized the principle has been accepted by economists and utilitarian philosophers whose judgment on the relation between material goods and utility or satisfaction carries weight. Thus Professor Alfred Marshall writes: “in accordance with a suggestion made by Daniel Bernoulli, we may perhaps suppose that the satisfaction which a person derives from his income may be regarded as beginning when he has enough to support life and afterwards as increasing by equal amounts with every equal successive percentage that is added to his income; and vice versa for loss of income.” The general principle is embodied in Bentham's utilitarian reasoning which has been widely accepted. The possibility of formulating the relation between feeling and its external cause is further supported by Fechner's investigations. This branch of Probabilities also obtains support from another part of the science, the calculation sanctioned by Laplace, of the disutility incident to error of measurement. Altogether it seems impossible to deny that some simple mathematical operations prescribed by the calculus of probabilities are sometimes serviceably employed to estimate prospective benefit in the subjective sense of desirable feeling.
20. Single Cases and “Series.”—Analogous to the question regarding the standard of belief which arose under a former head, a question regarding the standard of action arises under the head of expectation. The former question, it may be observed, arises chiefly with respect to events which are considered as singular, not forming part of a series. There is no doubt, there is a full belief, that if we go on tossing (unloaded) dice the event which consists of obtaining either a five or a six will occur in approximately of the trials. The important question is what is or should be our state of mind with regard to the result of a trial which is sui generis and not to be repeated, like the choice of a casket in the Merchant of Venice. A similar difficulty is presented by singular events, with respect to volition. Is the chance of one to a thousand of the prize £1000 at a lottery approximately equivalent to £1 in the eyes of a person who for once, and once only, has the offer of such a stake? The question is separable from one with which it is often confounded, the one discussed in the last paragraph what is the “moral” value of the prize? The person might be a millionaire for whom £1 and £1000 both belong to the category of small change. The stake and the prize might both be “moral.” The better opinion seems that apart from a system of transactions like that in which an insurance company undertakes, or at least a “cross-series” of the kind which seem largely to operate in ordinary life, expectations in which the risks are very different are no longer equateable. So De Morgan with regard to the “single case” (the solitary transaction in question) declares that the “mathematical expectation is not a sufficient approximation to the actual phenomenon of the mind when benefits depend upon very small probabilities; even when the fortune of the player forms no part of the consideration” [without making allowance for the difference between “moral” and mathematical probabilities]. So Condorcet, “If one considers a single man and a single event there can be no kind of equality” (between expectations with very different risks). It is only for the long run—lorsqu'on embrasse la suite indéfinie des évènements—that the rule is valid: To the same effect at greater length the logicians Dr Venn and von Kries. Some of the mathematical writers have much to learn from their logical critics on this and other questions relating to first principles.
Section II.—Calculation of Probability.
21. Object of the Section.—In the following calculations the principal object is to ascertain the number of cases favourable to an event in proportion to the total number of possible cases.
- Below, par. 152.
- Consider the equivalent of Laplace's second principle given at par 9, above, and his third principle quoted at par. 10.
- Above, par. 12.
- In the more familiar form; that (of two independently fluctuating quantities) the mean of the product is the product of the means (cf. Czuber, Theorie der Beobachtungsfehler, p. 133).
- Above, par. 6.
- These peculiarities afford some justification for Laplace's restriction of the term expectation to “goods.” As to the wider definition here adopted see below, par. 94 and par. 95, note.
- Each fortune referred to is divided by a proper parameter. See below, par. 69.
- Op. cit. liv. II. ch. xiii. No. 41. Cf. liv. II. ch. i. No. 2.
- Principles of Economics, book III., ch. vi. § 6, p. 209, ed. 4.
- Cf. below, par. 71.
- Some further references bearing on the subject are given in a paper by the present writer on the “Pure Theory of Taxation,” No. III. Economic Journ. (1897), vii. 550-551.
- Below, par. 131.
- Above, par. 14.
- Above, par. 5.
- Article on “Probabilities” (Encyc. Metrop.), § 40.
- Essai (1785), pp. 142 et seq.
- Logic of Chance, ch. vi. §§ 24-28.
- Wahrscheinlichkeitsrechnung, pp. 184 seq.
- The relations of recent logicians to the older mathematical writers on Probabilities may be illustrated by the relations of modern “historical” economists to their more abstract predecessors.
- Of the two properties which have been found to characterize probability (above, par. 5)—proportionate (1) number of (equally) favourable cases and (2) frequency of observed occurrence—the former especially pertain to the data and quaesita of this section.