414 



INDUCTION. 



cient to shake our belief in the law 

 itself. But between these two classes 

 of events there is an itjtennediate 

 class, consisting of what are commonly 

 termed Coincidences : in other words, 

 those combinations of chances which 

 present some peculiar and unexpected 

 regularity, assimilating them, in so 

 far, to the results of law. As if, for 

 example, in a lottery of a thousand 

 tickets, the numbers should be drawn 

 in the exact order of what are called 

 the natural numbers, i, 2, 3, &c. 

 We have still to consider the prin- 

 ciples of evidence applicable to this 

 case : whether there is any difference 

 between coincidences and ordinary 

 events in the amount of testimony 

 or other evidence necessary to render 

 them credible. 



It is certain that, on every rational 

 principle of expectation, a combina- 

 tion of this peculiar sort may be 

 expected quite as often as any other 

 given series of a thousand numbers ; 

 that with perfectly fair dice, sixes 

 will be thrown twice, thrice, or any 

 number of times in succession, quite 

 as often in a thousand or a million 

 throws, as any other succession of 

 numbers fixed upon beforehand ; and 

 that no judicious player would give 

 greater odds against the one series 

 than against the other. Notwith- 

 standing this, there is a general dis- 

 position 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 natxire has greater difficulty in 

 producing regular combinations 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 Melanf/es, 

 contends that regular combinations, 

 though equally probable according to 



the mathematical theory with any 

 others, are physically less probable. 

 He appeals to connuon 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 

 affirm, with the most positive convic- 

 tion, that the dice were false ? 



The common and natural impres- 

 sion is in favour of D'Alembert : the 

 regular series would be thought much 

 more unlikely than an irregular. But 

 this common impression is, I appre- 

 hend, merely grounded on the fact, 

 that scarcely anybody remembers to 

 have ever seen one of these peculiar 

 coincidences : the reason of which is 

 simply that no one's experience ex- 

 tends to anything 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 succession is i di\ided by 

 the tenth power of 36 ; in other words, 

 such a concurrence 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 

 t n throws had been fixed upon, it 

 would have been exactly as unlikely 

 that in any individual's experience 

 that particular succession had ever 

 occurred ; although 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 particular series of throws, 

 but between all regular and all irregu- 

 lar successions 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 un- 

 fairness in the dice, is unquestionably 

 true. But this arises from a totally 



