558 LAPLACE. 



this is curious, for the Treatise may be described as an abridge- 

 ment of Poisson's Recherches . . .SUV la Prob., and Poisson himself 

 refers to his memoir of 1830 ; so that it might have been expected 

 that some, if not all, of our conclusions would have presented 

 themselves to Galloway's attention. 



998. Laplace discusses in his pages 284 — 286 the following 

 problem. An urn contains a large number, n, of balls, some white, 

 and the rest black; at each drawing a ball is extracted and re- 

 placed by a black ball : required the probability that after r 

 drawings there will be x white balls in the urn. 



999. The remainder of the Chapter, forming pages 287 — 303, 

 is devoted to investigations arising from the following problem. 

 There are two urns, A and B, each containing n balls, some white 

 and the rest black ; there are on the whole as many white balls as 

 black balls. A ball is drawn out from each urn and put into the 

 other urn; and this operation is repeated r times. Required the 

 probability that there will then be x white balls in the urn A. 



This problem is formed on one which was originally given by 

 Daniel Bernoulli; see Arts. 417, 587, 807, 921. 



Let ^^. ,. denote the required probability; then Laplace obtains 

 the following equation: 



/ic + lV , 2x A, x\ , f^ x~ly 



This equation however is too difficult for exact solution, and so 

 Laplace mutilates it most unsparingly. He supposes n to be very 

 large, and he says that we have then approximately 



Q/Z^ If 1. CL Z rg ft 



