CHANCE. 79 



player's habits and his method of shuffling remain 

 the same. 



But if the number of shuffles is very large, the cards 

 which occupied the positions 123 before the first shuffle 

 may, after the last shuffle, occupy the positions 



123, 231, 312, 321, 132, 213, 

 and the probability of each of these six hypotheses is 

 clearly the same and equal to ^ ; and this is true what- 

 ever be the numbers A • • • A> which we do not know. 

 The great number of shuffles, that is to say, the com- 

 plexity of the causes, has produced uniformity. 



This would apply without change if there were more 

 than three cards, but even with three the demonstra- 

 tion would be complicated, so I will content myself 

 with giving it for two cards only. We have now only 

 two hypotheses 



12, 21, 



with the probabilities />i and /a = I -A • Assume that 

 there are n shuffles, and that I win a shilling if the 

 cards are finally in the initial order, and that I lose one 

 if they are finally reversed. Then my mathematical 

 expectation will be 



(A -A)" 



The difference pi-pi is certainly smaller than i, so 

 that if n is very large, the value of my expectation 

 will be nothing, and we do not require to know p^ 

 and A to know that the game is fair. 



Nevertheless there would be an exception if one of 

 the numbers /, and A was equal to i and the other to 

 nothing. It would then hold good no longer^ because 

 our original hypotheses would be too simple. 



What we have just seen applies not only to the 



