X 



276 d'alembert. 



D'Alembert falls back on Biiffon's own statement ; for he supposes 

 that a man has 100000 crowns, and that he stakes 50000 at an 

 equal game, and he says that this man's damage if he loses is 

 greater than his advantage if he gains ; jDuisque dans le premier 

 cas, il s'appauvrira de la moitie ; et que dans le second, il ne 

 s'enrichira que du tiers. 



502. If a person has the chance ■ of gaining x and the 



chance — - — of losing y, his expectation on the ordinary theory 



is ~ ^ . D'Alembert obtains this result himself on the ordi- 



nary principles ; but then he thinks another result, namely 



— , miofht also be obtained and defended. Let jz denote the 



sum which a man should give for the privilege of being placed 

 in the position stated. If he gains he receives x, so that as he 



paid z his balance is x — z. Thus — is the correspondincj 



expectation. If he loses, as he has already paid z he will have 

 to pay y — ^ additional, so that his total loss is y, and his con- 

 sequent expectation - — ^'^- . Then ^— — — is his total ex- 



p -^ q p + q 



pectation, which ought to be zero if z is the fair sum for him 



to pay. Thus z = ^ ^^ . It is almost superfluous to observe 



that the words which we have printed in Italics amount to as- 

 signing a new meaning to the problem. Thus D'Alembert gives 

 us not two discordant solutions of the sa77ie problem, but solu- 

 tions of two different problems. See his further remarks on his 

 page 283. 



503. D'Alembert objects to the common rule of multiplying 

 the value to be obtained by the probability of obtaining it in 

 order to determine the expectation. He thinks that the pro- 

 bability is the principal element, and the value to be obtained 

 is subordinate. He brings the following example as an objection 

 against the ordinary theory; but his meaning is scarcely intel- 

 ligible : 



