LAPLACE. 527 



strations are very imperfect. The memoir of Cauchy to which we 

 have referred, is very laborious and difficult, so that this portion 

 of the Theorie...des Proh. remains in an unsatisfactory state. 



969. We now arrive at Livre ii, which is entitled TMorie 

 Generate des Prohahilites. 



It will be understood that when we speak of any Chapter in 

 Laplace's work without further specification, we always mean a 

 Chapter of Livre IL 



The first Chapter is entitled Principes generaux de cette TMorie. 

 This occupies pages 179 — 188 ; it gives a brief statement, with 

 exemplification, of the first principles of the subject. 



970. The second Chapter is entitled De la ProbcibiliU des 

 ^venemens composes d'Svenemens simples dont les possihiliUs respec- 

 tives sont donnees. This occupies pages 189 — 27-i ; it contains the 

 solution of several problems in direct probability ; we will notice 

 them in order. 



971. The first problem is one connected with a lottery ; see 

 Arts. 291, 44^8, 4<d5, 775, 864, 910. 



The present discussion adds to what Laplace had formerly 

 given an approximate calculation. The French lottery was com- 

 posed of 90 numbers, 5 of which were drawn at a time. Laplace 

 shews that it is about an even chance that in 86 drawingfs all 

 the numbers will appear. This approximate calculation is an 

 example of the formula for AV given by Laplace on page 159 of 

 his work ; see Art. 966. 



We may remark that Laplace also makes use of a rougher ap- 

 proximation originally given by De Moivre ; see Art. 292. 



972. On his page 201 Laplace takes the problem of odd and 

 even; see Arts. 350, 865, 882. 



Laplace adds the following problem. Suppose that an urn con- 

 tains X white balls, and the same number of black balls ; an even 

 number of balls is to be drawn out : required the probability that 

 as many white balls as black balls will be drawn out. 



The whole number of cases is found to be 2"^"^ — 1, and the 



