GKOUNDS OF DISBELIEF. 445 



upon beforehand; and that no judicious player would give greater odds 

 against the one series than against the other. Notwithstanding this, there 

 is a general disposition to regard the one as much more improbable than 

 the other, and as requiring much stronger evidence to make it credible. 

 Such is the force of this impression, that it has led some thinkers to the 

 conclusion, that nature has greater difficulty in producing regular combi- 

 nations than irregular ones ; or in other words, that there is some general 

 tendency of things, some law, which prevents regular combinations from 

 occurring, or at least from occurring so often as others. Among these 

 thinkers may be numbered D'Alembert ; who, in an Essay on Probabilities 

 to be found in the fifth volume of his Melanges, contends that regular 

 combinations, though equally probable according to the mathematical theo- 

 ry with any others, are physically less probable. He appeals to common 

 sense, or, in other words, to common impressions ; saying, if dice thrown 

 repeatedly in our presence gave sixes every time, should we not, before the 

 number of throws had reached ten (not to speak of thousands of millions), 

 be ready to affii'm, with the most positive conviction, that the dice were 

 false? 



The common and natural impression is in favor of D'Alembert : the reg- 

 ular series would be thought much more unlikely than an irregular. But 

 this common impression is, I apprehend, merely grounded on the fact, that 

 scarcely any body remembers to have ever seen one of these peculiar coin- 

 cidences : the reason of which is simply that no one's experience extends to 

 any thing like the number of trials, within which that or any other given 

 combination of events can be expected to happen. The chance of sixes on 

 a single throw of two dice being ^, the chance of sixes ten times in suc- 

 cession is 1 divided by the tenth power of 36 ; in other words, such a con- 

 currence is only likely to happen once in 3,656,158,440,062,976 trials, a 

 number which no dice-player's experience comes up to a millionth part of. 

 But if, instead of sixes ten times, any other given succession of ten throws 

 had been fixed upon, it would have been exactly as unlikely that in any 

 individual's experience that particular succession had ever occurred; al- 

 though this does not seem equally improbable, because no one would be 

 likely to have remembered whether it had occurred or not, and because the 

 comparison is tacitly made, not between sixes ten times and any one par- 

 ticular series of throws, but between all regular and all irregular succes- 

 sions taken together. 



That (as D'Alembert says) if the succession of sixes was actually thrown 

 before our eyes, we should ascribe it not to chance, but to unfairness in the 

 dice, is unquestionably true. But this arises from a totally different prin- 

 ciple. We should then be considering, not the probability of the fact in 

 itself, but the comparative probability with which, when it is known to 

 have happened, it may be referred to one or to another cause. The regu- 

 lar sei'ies is not at all less likely than the irregular one to be brought about 

 by chance, but it is much more likely than the irregular one to bo pro- 

 duced by design; or by some general cause operating through the struc- 

 ture of the dice. It is the nature of casual combinations to produce a 

 repetition of the same event, as often and no oftener than any other series 

 of events. But it is the nature of general causes to reproduce, in the same 

 circumstances, always the same event. Common sense and science alike 

 dictate that, aU other things being the same, we should rather attribute the 

 effect to a cause which if real would be very likely to produce it, than to a 

 cause which would be very unlikely to produce it. According to Laplace's 



