CALCULATION OF CHANCES. 83 



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

 of approximation can practically be made to the neces- 

 sary precision. 



The question falls within Laplace's sixth principle, 

 of which, a short distance back, we gave the demon- 

 stration. The given fact, that is to say, the series of 

 coincidences, may have originated 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 coincidences 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 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 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 clearly be impossible to 

 estimate that probability with anything like numerical 

 precision. The counter-probability, however, that of 

 the accidental origin of the coincidence, dwindling so 



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