JA3IES BERNOULLI. 73 



125. Suppose then that we have an urn containing white balls 



and black balls, and that the ratio of the number of the former 



to the latter is known to he that of 3 to 2. We learn from the 



preceding result that if we make 25550 drawings of a single ball, 



replacing each ball after it is drawn, the odds are 1000 to 1 that 



31 29 



the white balls drawn lie between —- and — : of the whole num- 



50 oO 



ber drawn. This is the direct use of James Bernoulli's theorem. 



But he himself proposed to employ it inversely in a far more 



important way. Suppose that in the preceding illustration we 



do not know anything beforehand of the ratio of the white balls 



to the black ; but that we have made a larg-e number of drawings, 



and have obtained a white ball B times, and a black ball S times : 



then according to James Bernoulli we are to infer that the 



ratio of the white balls to the black balls in the urn is approxi- 



r) 



mately — . To determine the precise numerical estimate of the 



probability of this inference requires further investigation : we 

 shall find as we proceed that this has been done in two ways, 

 by an inversion of James Bernoulli's theorem, or by the aid of 

 another theorem called Bayes's theorem ; the results apj^roximately 

 agree. See Laplace, Theorie.,.des Proh.... pages 282 and 3CG. 



126. We have spoken of the inverse use of James Bernoulli's 

 theorem as the most important; and it is clear that he himself 

 was fully aware of this. This use of the theorem was that which 

 Leibnitz found it difficult to admit, and which James Bernoulli 

 maintained against him; seethe correspondence quoted in Art. 59, 

 pages 77, 83, 87, 94, 97. 



127. A memoir on infinite series follows the Ars Conjectandi, 

 and occupies pages 24)1 — 306 of the volume ; this is contained in 

 the collected edition of James Bernoulli's works, Geneva, 1744 : it 

 is there broken up into parts and distributed through the two 

 volumes of which the edition consists. 



This memoir is unconnected with our subject, and we will 

 therefore only briefly notice some points of interest which it 

 presents. 



