THE THEORY OF PROBABILITY. 245 



instance, that the probabilities would be the same when 

 coins are thrown successively as when thrown simul 

 taneously. a Some men of high ability, such as Ancillon, 

 Moses Mendelssohn, Garve, b Auguste Comte c and J. S. 

 Mill, d have so far misapprehended the theory, as to 

 question its value or even to dispute altogether its 

 vaKdity. 



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. 6 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 the most casual 

 event takes place the more likely it is to happen again; 

 because there is some slight empirical evidence of a 

 tendency, as will afterwards be pointed out. The source of 

 the fallacy is to be found entirely in the feelings of 

 surprise with which we witness an event happening by 

 apparent 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- 



a Todhunter, p. 279. b Ibid. p. 453. 



c Positive Philosophy/ translated by Martineau, vol. ii. p. 120. 

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

 e Montucla, Histoire/ vol. iii. p. 405. Todhunter, p. 263. 



