LAPIACE. 541 



which the chance is 1—p, and then a run of t. Thus the part 

 of z^ is 



'X 



Z 



^(1-i^)/^ or^/(l-^;)^,_,. 



We have said that Laplace adopts nearly the method we have 

 given ; but he is rather obscure. In the method we have given 

 P^ denotes the probability of the following compound event : no 

 run of i before the (ic — /— 4)^^ trial, the (a?— ^— 4)^^ trial un- 

 favourable, and then the next three trials favourable. Similarly 

 our Pg would denote the probability of the following compound 

 event; no run of i before the (a? — ^ — 2)^^ trial, the {x—i—Tf^ 

 trial unfavourable, and the next trial favourable. Laplace says, 

 Nommons P' la probability qu'il n'arrivera pas au coup x — i—% 

 Now Laplace does not formally say that there is to be no run of 

 i before the {x — i— 2)^^ trial ; but this must be understood. Then 

 his P agrees with our Pj if we omit the last of the three clauses 

 which form our account of the probability represented by Pg ; so 

 that in fact pP' with Laplace denotes the same as F^ with us. 



Laplace gives the integral of the equation (2), and finally ob- 

 tains the same result as we have exhibited in Art. 325. 



986. Laplace then proceeds to find the probability that one 

 of two players should have a run of i successes before the other ; 

 this investigation adds nothing to what Condorcet had given, but 

 is more commodious in form. Laplace's result on his page 250 

 will be found on examination to d^^t^o, with what we have griven 

 in Art. 680, after Condorc3t. 



Laplace then supplies some new matter, in which he considers 

 the expectation of each player supposing that after failiug he 

 deposits a franc, and that the sum of the deposits is taken by him 

 who first has a run of i successes. 



987. Laplace's next problem is the following. An urn con- 

 tains 71 + 1 balls marked respectively 0, 1, ... w ; a ball is drawn 

 and replaced : required the probability that after i drawings the 

 sum of the numbers drawn will be s. This problem and applica- 

 tions of it occupy pages 253 — 261. See Arts. 888, 915. 



The problem is due to De Moivre ; see Arts. 149, 364. La- 

 place's solution of the problem is very laborious. We will pass to 



