TREMBLEY. 417 



. / N /I \m (-, ^ ^ m(m — 1) x^ 



m (m — 1) (m — 2) x^ 



+ ^ 



where m =f+ 7i — 1. 



Now if X represents ^'s skill the probability that in 2n —f— h 

 games A would win 7i —f and B would win n — li is ic""-^ (1 — x)""'^, 

 disregarding a numerical coefficient which we do not want. 



Hence if A wins n —f games and B wins n — h, which is now 

 the observed event, we infer that the chance that A's skill is x is 



x""-^ (1 - x)''-"" dx 



f 



J a 



x^-f (1 _ xf-'' dx 



Therefore the fraction of the stake to which B is entitled is 



<f> {x) x''-' (1 - a?)"-' dx 



L 



x""-^ (1 - xy-"" dx 



All this involves only Laplace's ordinary theory. Now the 

 following is Trembley's method. Consider ^ (x) ; the first term 

 is (1 — xy ; this represents the chance that B will win m games 

 running on the supposition that his skill is 1 — x. If we do not 

 know his skill a ^7^iori we must substitute instead of (1 — a?)"* the 

 chance that B will win m games running, computed from the 

 observed fact that he has won 7i — h games to ^'s n —f games. 

 This chance is, by Art, 7Q6, 



Ui+f-l\2fi-f-h + l 



' ''"^ ^—j^r^r- = ^^ say. 



I — h [2n '^ 



Again consider the term rax (1 - xf'"^ in ix). This represents 

 the chance that B will win m - 1 games out of m, on the suppo- 

 sition that his skill is \-x. If we do not know his skill a jpriori 

 w^e must substitute instead of this the chance that B will win 



27 



