480 TREMBLEY. 



change. Hence Trembley's investigation of Peter's chance differs 

 from the method which we have exemplified in Art. 189. 



802. Trembley makes an attemjDt to solve the problem of 

 Her for three players ; but his solution is quite unsound. Sup- 

 pose there are three players, Paul, James, and Peter. Trembley 

 considers that the chances of Paul and James are in the propor- 

 tion of the chance of the first and second players when there are 

 only two players ; and he denotes these chances by x and y. He 

 takes aj to ?/ as 8496 to 8079 ; but these numbers are of no con- 

 sequence for our purpose. He supposes that the chances of James 

 and Peter are also in the same proportion. This would not be 

 quite accurate, because when James is estimating his chance with 

 respect to Peter he would have some knowledge of Paul's card ; 

 whereas in the case of Paul and James, the former had no know- 

 ledge of any other card than his own to guide him in retaining or 

 exchanging. 



But this is only a minute point. Trembley's error is in the 



next step. He considers that is the chance that Paul will 



x + y 



beat James, and that —^ — is the chance that Peter will beat 



x-\-y 



James ; he infers that -. — ^-^ is the chance that both Paul and 



{x-^yf 



Peter will beat James, so that James will be thrown out at the 

 first trial. This is false: the game is so constructed that the 



players are nearly on the same footing, so that - is very nearly 



o 



the chance that a given player will be excluded at the first trial. 



1 



Trembley's solution would give - as the chance that James will 



be excluded ii x=y) whereas -^ should then be the value. 



X 11 



The error arises from the fact that and — '- do not 



x-\- y ^ + y 



here represent independent chances ; of course if Paul has a higher 



card than James, this alone affords presumption that James will 



rather have a card inferior to that of Peter than superior. This 



error at the beginning vitiates Trembley's solution. 



