DANIEL BERXOULLI. 229 



employed the Differential Calculus, and the present memoir con- 

 tains remarks which would serve to explain the process of Art. 402 ; 

 but the remarks are such as any student could easily supply 

 for himself We shall see the point illustrated in another memoir. 

 See Art. 417. 



410. The problem which Daniel Bernoulli solves is in its 

 simplest form as follows : In a bag are 2n cards ; two of them are 

 marked 1, two of them are marked 2, two of them are marked 3, ... 

 and so on. We draw out m cards ; required the probable number 

 oi pairs which remain in the bag. 



We give the solution of Daniel Bernoulli with some changes of 

 notation. Suppose that a?,,, pairs remain after m cards have been 

 drawn out ; let a new drawing be made. The card thus drawn out 

 is either one of the cards of a pair, or it is not ; the probabilities 

 for these two cases are proportional to 20?,,^, and 2n — 2x,,^ — m re- 

 spectively : in the former case there remain x^^ — 1 pairs in the bag, 

 and in the latter case there remain x,^ pairs. Thus by ordinary 

 principles 



^ ^^m G^., - 1) + (2>i - 2^,^ - m) x^ 

 "^1 2/1 - m 



2n — m— 2 



Zn — m 



iC„j 



We can thus form in succession x^, x^, ^3> ••• As x^=n we 

 find that 



(2)1 — on) (2n - m — 1) 



•^ni "~ 



2 {2n - 1) 



411. The problem is extended by Daniel Bernoulli afterwards 

 to a greater generality ; but we have given sufficient to enable the 

 reader to understand the nature of the present memoir, and of that 

 to which we now proceed. 



412. The next memoir is entitled De duratione media matri- 

 moniorum, pro qitacunque conjugum aetate, aliisque quaestionihus 

 affinibus. 



This memoir is closely connected with the preceding ; it fol- 

 lows in the same volume of the JS'ovi Comm...Petrop., and occupies 

 pages 99—126. 



