380 CONDOHCET. 



Laving iV is ^ . In fact the hypothesis leads to the same conclu- 



sion as we should obtain from the hypothesis that A and N are 

 always equally likely to occur. 



In his first problem Condorcet assumes that the probability of 

 each event remains constant during the observations ; in his second 

 problem he says that he does not assume this. But we must 

 observe that to abstain from assuming that an element is constant 

 is different from distinctly assuming that it is not constant. Con- 

 dorcet, as we shall see, seems to confound these two things. His 

 second problem does not exclude the case of a constant probability, 

 for as we have remarked it is coincident with the case in which 



there is a constant probability equal to ^ . 



The introduction of this second problem, and of others similar 

 to it is peculiar to Condorcet. We shall immediately see an appli- 

 cation which he makes of the novelty in his third problem ; and we 

 shall not be able to commend it. 



699. Condorcet's third problem is thus enunciated : 



On suppose dans ce probleme que Ton ignore si a chaque fois la pro- 

 bahilite d' avoir A qvl N reste la meme, on si elle varie a chaque fois, de 

 nianiere quelle puisse avoir une valeur quelconque depuis zero jusqu'a 

 r unite, et Ton demande, sacliant que Ton a eu m evenemens -4, et n 

 evenemens N^ quelle est la probabilite d'amener A ou -^V. 



The following is Condorcet's solution. If the probability is 



constant, then the probability of obtaining m occurrences of A 



I m -\- n r^ 

 and 71 occurrences of N is ', , - x"' (1 — xY dx, that is 



If the probability is not constant, then, as in 



\m \n \m-\-7i + l 



the second problem, the probability of obtaining 7n occurrences of ^ 



I !ii -\-n \ 

 and n occurrences of N is ^r^^rr^ . Hence he infers that the 



P Q 



probabilities of the hypothesis are respectively and ^ , 



\m\n 1 



where P= — — — ^ and Q = 



m-]-n + l 2 



m+n 



