OF THE CALCULATION OF CHANCES. 385 



themselves. The probabilities of life at different ages or in different cli- 

 mates ; the probabilities of recovery from a particular disease ; the chances 

 of the biith of male or female offspring ; the chances of the destruction of 

 houses or other property by fire; the chances of the loss of a ship in a 

 particular voyage, are deduced from bills of mortality, returns from hos- 

 pitals, registei's of births, of shipwrecks, etc., that is, from the observ-ed 

 frequency not of the causes, but of the effects. The reason is, that in all 

 these classes of facts the causes are either not amenable to direct observa- 

 tion at all, or not with the requisite precision, and we have no means of 

 judging of their frequency except from the empirical law afforded by the 

 frequency of the effects. The inference does not the less depend on cau- 

 sation alone. We reason from an effect to a similar effect by passing 

 through the cause. If the actuary of an insurance office infers from his 

 tables that among a hundred persons now living of a particular age, five 

 on the average will attain the age of seventy, his inference is legitimate, 

 not for the simple reason that this is the proportion who have lived till 

 seventy in times past, but because the fact of their having so lived shows 

 that this is the proportion existing, at that place and time, between the 

 causes which prolong life to the age of seventy and those tending to bring 

 it to an earlier close.* 



§ 5. From the preceding principles it is easy to deduce the demonstra- 

 tion of that theorem of the doctrine of probabilities which is the founda- 

 tion of its application to inquiries for ascertaining the occurrence of a 

 given event, or the reality of an individual fact. The signs or evidences by 

 which a fact is usually proved are some of its consequences ; and the in- 

 quiry hinges upon determining what cause is most likely to have produced 

 a given effect. The theorem applicable to such investigations is the Sixth 

 Principle in Laplace's "Essai PhilosopJiique sur les Probabilites^'' which is 

 described by him as the " fundamental principle of that branch of the Analy- 

 sis of Chances which consists in ascending from events to their causes."f 



Given an effect to be accounted for, and there being several causes which 

 might have produced it, but of the presence of which in the particular case 

 nothing is known ; the probability that the effect -was produced by any one 

 of these causes is as the antecedent probability of the cause, multiplied by 

 the probability that the cause, if it existed, toould have produced the given 

 effect. 



Let M be the effect, and A,B, two causes, by either of which it might 



* The writer last quoted says that the valuation of chances bj' comparing the number of 

 cases in which the event occurs with the number in which it does not occur, "would gen- 

 erally be wholly erroneous," and "is not the true theory of probability." It is at least that 

 which forms the foundation of insurance, and of all those calculations of chances in the busi- 

 ness of life which experience so abundantly verifies. The reason which the reviewer gives for 

 rejecting the theory is, that it " would regard an event as certain which had hitherto never 

 failed ; which is exceedingly far from the truth, even for a very large number of constant suc- 

 cesses." This is not a defect in a particular theory, but in any theory of chances. No prin- 

 ciple of evaluation can provide for such a case as that which the reviewer supposes. If an 

 event has never once failed, in a number of trials sufficient to eliminate chance, it really has 

 all the certainty which can be given by an empirical law ; it is certain during tlie continuance 

 of the same collocation of causes which existed during the observations. If it ever foils, it is 

 in consequence of some change in that collocation. Now, no theory of chances will enable us 

 to infer the future probability of an event from tlie past, if the causes in operation, capable of 

 influencing the event, have inteiTnediately undergone a change. 



t Pp. 18, 19. The theorem is not stated by Laplace in the exact terms in which I have 

 stated it ; but the identity of import of the two modes of expression is easily demonstrable. 



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