LAPLACE. 467 



are drawn of which p are white and q are black : required the pro- 

 bability of drawing m white tickets and n black tickets in the next 

 m + n drawings. 



Laplace gives for the required probability 



/, 



a^^'" (1 _ xY'-'' clx 



j x" (1 -xydx 



'^ 



so that of course the m white tickets and n black tickets are sup- 

 posed to be draT^^l in an assigned order ; see Arts. ^0^, 76G, 843. 

 Laplace effects the integration, and approximates by the aid of a 

 formula which he takes from Euler, and which we usually call 

 Stirling's Theorem. 



The problem here considered is not explicitly reproduced in the 

 Theorie. . .des Proh., though it is involved in the Chapter which forms 

 pages 363—401. 



871. After discussing this problem Laplace says, 



La solution de ce Probleme donne une methode directe pour deter- 

 miner la probabilite des evenemens futurs d'apres ceux qui sont deja 

 arrives ; mais cette matiere etant fort etendue, je me bornerai ici a 

 donner une demonstration assez singuliere du theoreme suivant. 



On peut suppose)' les nomhres p e^ q tellement grands, qiuil devienne 

 aussi ajJjirochant que Von voudra de la certitude, que le rapport du 

 nomhre de billets blancs au nomhre total des billets renfermes dans 



Vurne, est compris e^itre les deux limites — w, et — 1- w, to pouvant 



p + qp-fq-^ 



etre suj^j^ose moindre quaucune grandeur donnee. 



The probability of the ratio lying between the specified limits is 



x^{l -xydx 



[ 



[ x^ (1 - xy dx 



^ 



where the inteoral in the numerator is to be taken between the 



limits — w and — V «. Laplace by a rude process of 



^ + ^ p-\-q 



30—2 



