OF THE CALCULATION OF CHANCES. 75 



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 ; being altogether once in thirty-six times. 

 The chance of the same cast three times successively is, by 



a similar reasoning, - 3 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 correctly 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 originated either in a casual conjunc- 

 tion 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 coinci- 

 dences. The probabilities, therefore, that the coincidences 

 are produced by the two causes in question, are as the ante- 

 cedent probabilities of the causes. One of these, the ante- 

 cedent 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 nature 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 



