LAGRANGE. 31o 



chance that the event shall happen just h times ; he starts from 

 the same equation and by a different determination of the arbi- 

 trary quantities arrives at the result which is well known, 

 namely, 



y (1 -pr' \a_ 



\b \a — h 



Lagrange refers to De Moivre, page 15, for one solution, and 

 adds : mais celle que nous venons d'en donner est non seulement 

 plus simple, mais elle a de plus I'avantage d'etre d^duite de prin- 

 cipes directs. 



But it should be observed that De Moivre solves the problem 

 again on his page 27; and here he indicates the modern method, 

 which is self-evident. See Art. 257. 



It seems curious for Lagrange to speak of his method as more 

 simple than De Moivre's, seeing it involves an elaborate solution 

 of an equation in Finite Differences. 



580. Lagrange's second problem is the following : 



On sujDpose qu'a chaqiie coup il puisse arriver deux 6venemens dont 

 les probabilites respectives soient p et q; et on demande le sort d'un 

 joueur qui parieroit d'amener le premier de ces evenemens h fois au 

 moins et le second c fois au moins, en un nombre a de coups. 



The enunciation does not state distinctly what the suppositions 

 really are, namely that at every trial either the first event happens, 

 or the second, or neither of them ; these three cases are mutually 

 exclusive, so that the probability of the last at a single trial 

 is 1 —p — q. It is a good problem, well solved ; the solution is 

 presented in a more elementary shape by Trembley in a memoir 

 which we shall hereafter notice. 



581. The third problem is the following : 



Les memes choses etant supposees que daus le Probleme li, on de- 

 mande le sort d'un joueur qui parieroit d'amener, dans un nombre de 

 coups indetermine, le second des deux Evenemens h fois avant que le 

 premier fut arrive a fois. 



Let 7/^ J be the chance of the player when he has to obtain the 

 second event t times before the first event occurs x times. Then 



