d'alembert. 281 



D'Alembert brings forward four remarks which shew that thi^ 

 mode of explaining the difficulty is unsatisfactory. One of theoi 

 is the following : instead of supposing that one crown is to be 

 received for head at the first throw, two for head at the second 

 throw, four for head at the third throw, and so on, suppose that in 

 each case only one crown is to be received. Then, although theo- 

 retically the game may endure to infinity, yet the value of the 

 expectation is finite. This remark may be said to contradict a 

 conclusion at which D'Alembert arrived in his article Croix oil 

 Pile, which we noticed in Ai"t. ^^d. 



51-i. The case just brought forward is interesting because 

 D'Alembert admits that it might supply an objection to his prin- 

 ciples. He tries to repel the objection by saying that it only leads 

 him to suspect another principle of the ordinary theory, namely 

 that in virtue of which the total expectation is taken to be equal 

 to the sum of the partial expectations ; see his pages 299 — 301. 



* 



515. D'Alembert thus sums up his objections against the 

 ordinary theory : 



Pour resumer en un mot tons ines doutes sur le calcul des pro- 

 babilites, et les mettre sous les yeux des vrais Juges; voici ce que 

 j'accorde et ce que je nie dans les raisonnemens explicites ou implicites 

 sur lesquels ce calcul me paroit fonde. 



Premier raisonnement. Le u ombre des combinaisons qui amenent 

 tel cas, est au nombre des combinaisons qui amenent tel autre cas, 

 comme p est ^ q. Je conviens de cette verite qui est purement ma- 

 thematique; done, conclut-on, la probabilite du premier cas est a celle 

 du second comme j^ est a q. Yoila ce que je nie, ou du moins de 

 quoi je doute fort; et je crois que si, par exemple, ^^ = 5', et que dans 

 le second cas Je meme evenement se trouve un tres-grand nombre de 

 fois de suite, il sera moins probable physiquement que le premier, 

 quoique les probabilites mathematiques soient egales. 



Second raisonnement. LaprobabiUte - est a la probabilite — comme 



^ m n 



np ecus est a mp ecus. J'en conviens; done— x mp ecus = - x np ecus; 



j'en conviens encore; done Vesperance, ou ce qui est la meme chose, 



