01^ THE CALCULATION OF CHANCES. 



357 



§ 5. From the preceding principles 

 it is easy to deduce the demonstration 

 of that theorem of the doctrine of pro- 

 babilities which is the foundation of 

 its application to inquiries for ascer- 

 taining 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 inquiry 

 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 Bssai Philoso- 

 phique sur les Probabilites, which is 

 described by him as the "fundamental 

 principle of that branch of the Analy- 

 sis of Chances which consists in as- 

 cending from events to their causes." * 



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 proba- 

 bility that the effect was produced 

 by any one of these causes is as the 

 antecedent probability of the cause, 

 viuUiplied by the probability that the 



tion of insurance, and of all those calcula- 

 tions of chances in the business of life 

 which experience so abtmdantly verifies. 

 The reason which the reviewer gives for 

 rejecting the theory, is that it " would re- 

 gard an event as certain which had hitherto 

 never failed ; which is exceedingly f;ir from 

 the truth, even for a very large number of 

 constant successes." This is not a defect 

 in a particular theory, but in any theory 

 of chances. No principle 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 suflBcient 

 to eliminate chance, it really has all the 

 certainty which can be given by an em- 

 pirical law : it is certain during the con- 

 tinuance of the same collocation of causes 

 which existed during the observations. If 

 it ever fails, 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 the past, if 

 the causes in operation capable of influen- 

 cing the event have internaediately under- 

 gone a change, 



* Pp. 18, 19. The theorem is not stated by 

 Laplace in the exact terms in which I have 

 Slated it ; but the identity of import of the 

 two modes of expression is easily demon- 

 strable. 



Cause, if it existed, would have pro- 

 duced the given effect. 



Let M be the effect, and A, B, two 

 causes, by either of which it might 

 have been prodiiced. To find the 

 probability that it was produced by 

 the one and not by the other, ascer- 

 tain which of the two is most likely 

 to have existed, and which of them, 

 if it did exist, was most likely to pro- 

 duce the effect M : the probability 

 sought is a compound of these two 

 probabilities. 



Cask I. Let the causes be both 

 alike in the second respect ; either A 

 or B, when it exists, being supposed 

 equally likely (or equally certain) to 

 produce M ; but let A be in itself 

 twice as likely as B to exist, that is, 

 twice as frequent a phenomenon. 

 Then it is twice as likely to have 

 existed in this case, and to have been 

 the cause which produced M. 



For, since A exists in nature twice 

 as often as B, in any 300 cases in 

 which one or other existed, A has 

 existed 200 times and B 100. But 

 either A or B must have existed 

 wherever M is produced : therefore 

 in 300 times that M is produced, A 

 was the producing cause 200 times, 

 B only 100, that is, in the ratio of 2 

 to I. Thus, then, if the causes are 

 alike in their capacity of producing 

 the effect, the probability as to which 

 actually produced it is in the ratio 

 of their antecedent probabilities. 



Case IL Reversing the last hypo- 

 thesis, let us suppose that the causes 

 are equally frequent, equally likely to 

 have existed, but not equally likely, 

 if they did exist, to produce Si : that 

 in three times in which A occurs, it 

 produces that effect twice, while B, 

 in three times, produces it only once. 

 Since the two causes are equally fre- 

 quent in their occurrence ; in every 

 six times that either one or the other 

 exists, A exists three times and B 

 three times. A, of its three times, 

 produces M in two ; B, of its three 

 times, produces M in one. Thiis, in 

 the whole six times, M is only pro- 

 duced thrice ; but of that thrice it is 



