446 INDUCTION. 



sixth theorem, which we demonstrated in a former chapter, the difference 

 of probability arising from the superior efficacy of the constant cause, un- 

 fairness in the dice, would after a very few throws far outweigh any ante- 

 cedent probability which there could be against its existence. 



D'Alembert should have put the question in another manner. He should 

 have supposed that we had ourselves previously tried the dice, and knew 

 by ample experience that they were fair. Another person then tries them 

 in our absence, and assures us that he threw sixes ten times in succession. 

 Is the assertion credible or not? Here the effect to be accounted for is 

 not the occurrence itself, but the fact of the witness's asserting it. This 

 may arise either from its having really happened, or from some other 

 cause. What we have to estimate is the comparative probability of these 

 two suppositions. 



If the witness affirmed that he had thrown any other series of numbers, 

 supposing him to be a person of veracity, and tolerable accuracy, and to 

 profess that he took particular notice, we should believe him. But the 

 ten sixes are exactly as likely to have been really thrown as the other se- 

 ries. If, therefore, this assertion is less credible than the other, the reason 

 must be, not that it is less likely than the other to be made truly, but that 

 it is more likely than the other to be made falsely. 



One reason obviously presents itself why what is called a coincidence, 

 should be oftener asserted falsely than an ordinary combination. It ex- 

 cites wonder. It gratifies the love of the marvelous. The motives, there- 

 fore, to falsehood, one of the most frequent of which is the desire to aston- 

 ish, operate more strongly in favor of this kind of assertion than of the 

 other kind. Thus far there is evidently more reason for discrediting an 

 alleged coincidence, than a statement in itself not more probable, but 

 which if made would not be thought remarkable. There are cases, how- 

 ever, in which the presumption on this ground would be the other way. 

 There are some witnesses who, the more extraordinary an occurrence 

 might appear, would be the more anxious to verify it by the utmost care- 

 fulness of observation before they would venture to believe it, and still 

 more before they would assert it to others. 



§ 6. Independently, however, of any peculiar chances of mendacity aris- 

 ing from the nature of the assertion, Laplace contends, that merely on the 

 general ground of .the fallibility of testimony, a coincidence is not credible 

 on the same amount of testimony on which we should be M'arranted in be- 

 lieving an ordinary combination of events. In order to do justice to his 

 argument, it is necessary to illustrate it by the example chosen by himself. 



If, says Laplace, there were one thousand tickets in a box, and one only 

 has been drawn out, then if an eye-witness affirms that the number drawn 

 was 79, this, though the chances were 999 in 1000 against it, is not on that 

 account the less credible ; its credibility is equal to the antecedent proba- 

 bility of the witness's veracity. But if there were in the box 999 black 

 balls and only one white, and the witness affirms that the white ball was 

 drawn, the case according to Laplace is very different : the credibility of 

 his assertion is but a small fraction of what it was in the former case ; the 

 reason of the difference being as follows : 



The witnesses of whom we are speaking must, from the nature of the 

 case, be of a kind whose credibility falls materially short of certainty; let 

 us suppose, then, the credibility of the witness in the case in question to 

 be -^ ; that is, let us suppose that in every ten statements which the wit- 



