D ALEMBEllT. 279 



of the remarks which have akeady been noticed. D'Alembei-t 

 records the origin of his doubts in these words : 



II y a pres de trente ans que j'avois forme ces doutes en Hsaiit 

 Texcellent livre de M. BernoulU de Arte conjectandi ; . . . 



He seems to have returned to his old error respecting Croix 

 ou Pile with fresh ardour : he says, 



...si les trois cas, croix, pile et croix, pile et 2^ile, les seuls qui 

 puissent arriver dans le jeu propose, ne sont pas egalement possibles, 

 ce n'est point, ce me semble, par la raison qu'on en apporte commu- 



nement, que la probabilite du premier est - , et celle des deux autres 



Q X - ou - . Plus j'y pense, et plus il me paroit que Tiiathematique- 

 nient parlaut, ces trois coups sont egalement possibles... 



510. D'Alembert introduces another point in which he ob- 

 jects to a principle commonly received. He will not admit that 

 it is the same thing to toss one coin m times in succession, or 

 to toss m coins simultaneously. He says it is perhaps physically 

 speaking more possible to have the same face occurring simul- 

 taneously an assigned number of times with m coins tossed at 

 once, than to have the same face repeated the same assigned 

 number of times when one coin is tossed ')n times. But no person 

 will allow what D'Alembert states. We can indeed suppose circum- 

 stances in which the two cases are not quite the same ; for example 

 if the coins used are not perfectly symmetrical, so that they 

 have a tendency to fall on one face rather than on the other. 

 But we should in such a case expect a run of resemblances rather 

 in using one coin for m throws, than in using m coins at once. 

 Take for a simple example m — 2. We should have rather more 



than -r as the chance for the former result, and only - lor tlie 



latter; see Laplace, Theorie...des Proh. page 402. 



511. D'Alembert says on his page 290, II y a quelque temps 

 qu'un Joueur me demanda en combien de coups consecutifs on 

 pouvoit parier avec avantage d'amener une face donnee d'un de — 

 This is the old question proposed to Pascal by the Chevalier de 



