OF THE CALCULATION OF CHANCES. 



3S9 



of approximation can practically be 

 made to the necessary precision. 



The question falls within Laplace's 

 Sixth Principle, just demonstrated. 

 The given fact, that is to say, the 

 series of coincidences, may have origi- 

 nated either in a casual conjunction 

 of causes or in a law of nature. The 

 probabilities, therefore, that the fact 

 originated in these two modes are, as 

 their antecedent probabilities, multi- 

 plied by the probabilities that if they 

 existed they would produce the effect. 

 But the particular combination of 

 chances, if it occurred, or the law of 

 nature if real, would certainly pro- 

 duce the series of coincidences. The 

 probabilities, therefore, that the co- 

 incidences are produced by the two 

 causes in question are as the ante- 

 cedent probabilities of the causes. 

 One of these, the antecedent probabi- 

 lity of the combination of mere chances 

 which would produce the given result, 

 is an appreciable quantity. The an- 

 tecedent probability of the other sup- 

 position may be susceptible of a more 

 or less exact estimation, according to 

 the nature of the case. 



In some cases the coincidence, sup- 

 posing it to be the result of causation 

 at all, must be the result of a known 

 cause, as the succession of aces, if not 

 accidental, must arise from the load- 

 ing of the die. In such cases we 

 may be able to form a conjecture as 

 to the antecedent probability of such 

 a circumstance from the characters of 

 the parties concerned, or other such 

 evidence ; but it would be impossible 

 to estimate that probability with any- 

 thing like numerical precision. The 

 counter-probability, however, that of 

 the accidental origin of the coinci- 

 dence, dwindling so rapidly as it does 

 at each new trial ; the stage is soon 

 reached at which the chance of un- 

 fairness in the die, however small in 

 itself, must be greater than that of 

 a casual coincidence ; and on this 

 ground a practical decision can gene- 

 rally be come to without much hesita- 

 tion, if there be the power of repeating 

 the experiment. 



When, however, the coincidence is 

 one which cannot be accounted for 

 by any known cause, and the con- 

 nection between the two phenomena,, 

 if produced by causation, must be the 

 result of some law of nature hitherto 

 unknown, which is the case we had 

 in view in the last chapter ; then, 

 though the probability of a casual 

 coincidence may be capable of ap- 

 preciation, that of the counter-sup- 

 position, the existence of an undis- 

 covered law of nature, is clearl}' un- 

 susceptible of even an approximate 

 valuation. In order to have the data 

 which such a case would require, it 

 would be necessary to know what 

 proportion of all the individual se- 

 quences or co-existences occurring 

 in nature are the result of law, and 

 what proportion are mere casual coin- 

 cidences. It being evident that we 

 cannot form any plausible conjecture 

 as to this proportion, much less ap- 

 preciate it numerically, we cannot 

 attempt any precise estimation of the 

 comparative probabilities. But of this 

 we are sure, that the detection of 

 an unknown law of nature — of some 

 previously unrecognised constancy of 

 conjunction among phenomena — is no 

 uncommon event. If, therefore, the 

 number of instances in which a coin- 

 cidence is observed, over and above 

 that which would arise on the average 

 from the mere concurrence of chances, 

 be such that so great an amount of co- 

 incidences from accident alone would 

 be an extremely uncommon event ; 

 we have reason to conclude that the 

 coincidence is the effect of causation, 

 and may be received (subject to cor- 

 rection from further experience) as an 

 empirical law. Further than this, in 

 point of precision, we cannot go ; nor, 

 in most cases, is greater precision re- 

 quired for the solution of any practi- 

 cal doubt.* 



* For a fuller treatment of the many 

 interesting questions raised by the theory 

 of probabilities, I may now refer to a recent 

 work by Mr. Venn, Fellow of Caius College, 

 Cambridge, "The Logic of Chance," one 

 of the most thoughtful and philosophical 

 treatises on any subject connected with 



