CALCULATION OF CHANCES. 325 



§ 4. From the preceding view of the foundation of the doctrine of 

 chances, its gcncnd principlt^:? may bo seen to be appHcable in a rough 

 way to many subjects which are by no means amenable to its precise 

 calculations. To ivnder these applicable, there must be numerical 

 data, derived from the observation of a vei-y large number of instances. 

 The probabilities of life at dillerent ages, or in different climates; the 

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

 birtli of male or female offspring ; the chances of the loss of a vessel 

 in a particular voyage ; all these admit of estimation sufficiently pre- 

 cise to render the numerical appreciation of their amount a thing of 

 pnictical value ; because there are bills of mortality, returns from 

 hospitals, registers of birtlis, of shipwrecks, &c., founded on cases 

 sufficiently numerous to afford average proportions which do not 

 materially vaiy from year to year, or from ten yeai's to ten years. But 

 where obser\'ation and experiment have not afforded a set of instances 

 sufficiently numerous to eliminate chance, and sufficiently various to 

 eliminate all non-essential specialities of circumstance, to attempt to 

 calcidate chances is to convert mere ignorance into dangerous error 

 by clothing it in the garb of knowledge. 



It remains to examine the bearing of the doctrine of chances upon 

 the peculiar problem for the sake of which wo have on this occasion 

 adverted to it, namely, how to distinguish coincidences wliich are 

 casual from those which are the result of law ; from those in which the 

 facts which accompany or follow one another are somehow connected 

 through causation. 



§ o. The doctrine of chances affords means by which, if we knew 

 the average number of coincidences to be looked for between two 

 phenomena connected only casually, we could determine how often 

 any given deviation from that average will occur by chance. If the 



probability of any casual coincidence, considered in itself, be — , the 



m 

 probability that the same coincidence will be repeated n times in suc- 

 cession is — ^. For example, in one throw of a die the probability of 



ace being — ; the probability of' throwing ace twice in succession will 



. . 1 



be 1 divided by the square of 6, or — , For ace is thrown at the first 



throw once in six, or six in thirty-six times ; and of those six, the die 

 being cast again, ace will be thrown but once j being altogether Once 

 in thirty-six times. Tie chance of the same cast three times succes- 

 sively ia, by a similar reasoning, —^ or — — : that is, the event will hap- 

 pen, on a lai'ge average, only once in two hundred and sixteen 

 throws. 



We have thus a rule by which to estimate the probability that any- 

 given series of coincidences arises from chance ; provided we can 

 measure coiTectly the probability of a single coincidence. If we could 

 obtain an equally })reci8e expression for the probability that the same 

 series of coincidences arises from causation, wo should only have to 

 compare the numbers. This, however, can rarely be done. Let us 



