OF THE CALCULATION OF CHANCES. 387 



the peculiar problem which occupied us in the preceding chapter, namely, 

 how to distinguisli coincidences which are casual from those Avhich are the 

 result of law ; f I'ora those in which the facts which accompany or follow 

 one another are somehow connected through causation. 



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 probability that the same coincidence will 



be repeated n times in succession is — . For example, in one throw of a 



die the probability of ace being -; the probability of throwing ace twice 



in succession will be 1 divided by the square of 6, or — . For ace is thrown 



36 



at the fii'st 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 ; being altogether once in 



thirtj'-six times. The chance of the same cast three times successively is, 



by a similar reasoning, —^ or — — ; that is, the event will happen, on a large 



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 correct- 

 ly the probability of a single coincidence. If we can obtain an equally 

 precise expression for the probability that the same series of coincidences 

 arises from causation, we should only have to compare the numbers. This, 

 however, can rarely be done. Let us see what degree 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, multiplied 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 produce 

 the series of coincidences. The probabilities, therefore, that the coinci- 

 dences are produced by the two causes in question are as the antecedent 

 probabilities of the causes. One of these, the antecedent probability of the 

 combination of mere chances which would produce the given result, is an 

 appreciable quantity. The antecedent probability of the other supposition 

 may be susceptible of a more or less exact estimation, according to the na- 

 ture of the case. 



In some cases, the coincidence, supposing 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 loading of the die. In such cases we may 

 be able to form a conjecture as to the antecedent probability of such a cir- 

 cumstance from the characters of the parties concerned, or other such evi- 

 dence ; 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 coincidence, dwindling so rapidly as it does at 

 each new trial, the stage is soon reached at which the chance of unfairness 



