214 THE PRINCIPLES OF SCIENCE. [CHAP. 



coins are thrown successively as when thrown simul- 

 taneously. Some men of great reputation, such as 

 Ancillon, Moses Mendelssohn, Garve, Auguste Comte, 1 

 Poiusot, and J. S. Mill,* have so far misapprehended the 

 theory, as to question its value or even to dispute its 

 validity. The erroneous statements about the theory given 

 in the earlier editions of Mill's System of Logic were par- 

 tially withdrawn in the later editions. 



Many persons have a fallacious tendency to believe that 

 when a chance event has happened several times together 

 in an unusual conjunction, it is less likely to happen 

 again. D'Alembert seriously held that if head was thrown 

 three times running with a coin, tail would more probably 

 appear at the next trial. 3 Bequelin adopted the same 

 opinion, and yet there is no reason for it whatever. If 

 the event be really casual, what has gone before cannot in 

 the slightest degree influence it. As a matter of fact, the 

 more often a casual event takes place the more likely it is 

 to happen again ; because there is some slight empirical 

 evidence of a tendency. The source of the fallacy is to be 

 found entirely in the feelings of surprise with which we 

 witness an event happening by chance, in a manner which 

 seems to proceed from design. 



Misapprehension may also arise from overlooking the 

 difference between permutations and combinations. To 

 throw ten heads in succession with a coin is no more 

 unlikely than to throw any other particular succession 

 of heads and tails, but it is much less likely than five 

 heads and five tails without regard to their order, be- 

 cause there are no less than 252 different particular 

 throws which will give this result, when we abstract 

 the difference of order. 



Difficulties arise in the application of the theory from 

 our habitual disregard of slight probabilities. We are 

 obliged practically to accept truths as certain which are 

 nearly so, because it ceases to be worth while to calculate 

 the difference. No punishment could be inflicted if 

 absolutely certain evidence of guilt were required, and as 



1 Positive Philosophy, translated by Martineau, vol. ii. p. 120. 



2 System of Logic, bk. iii. chap. 18, 5th Ed. vol. ii. p. 61. 

 z Montucla, Histoire, vol. iii. p. 405 ; Todhunter, p. 263. 



