CONDORCET. 879 



same ; the real hypothesis involves much more, namely, that the 

 probability is of unknown value, any value between zero and unity 

 being equally likely a priori. 



Similarly we have the following result. Suppose the event A 

 has occurred m times and the event N has occurred n times ; sup- 

 pose that the probability of the two events is constantly the same, 

 but of unknown value, any value between a and h being equally 

 likely a priori ; required the probability that the probability of A 

 lies between certain limits a and ^ which are themselves com- 

 prised between a and h. 



The required probability is 



i 



' x'^il-xYdx 



f 



J a 



x"^ (1 - xy dx 



a 



Laplace sometimes speaks of such a result as the jyrohahilitii 

 that the p)Ossihilitij oi A lies between a and /3 ; see Theorie...des 

 Proh. Livre ii. Chapitre vi. See also De Morgan, Theory of Proba- 

 bilities, in the Encyclopcedia Metropolitana, Art. 77, and Essay on 

 Probahilities in the Cabinet Cyclopedia, page 87. 



698. Condorcet's second problem is thus enunciated : 



On suj^pose dans ce Problem e, que la probabilite de A et de N n'est 

 pas la meine dans tous les evenemens, mais qu'elle pent avoir pour 

 chacun une valeur quelconque depuis zero jusqu'a I'unite. 



Condorcet's solution depends essentially on this statement. The 

 probability of m occurrences of A, and n occurrences of N is 



\m-\-n ( r^ ) "' f f ^ ] " \m-\-n \ 



. I \ xdx \ \\ {1-x) dx\ , that is '. , -^^^i^Tn • 

 \]]}[^ [Jo ) Ih J lull!}. ^ 



The probability of having A again, after A has occurred m times 



and N has occurred n times, is found by changing the exponent m 



into m + 1, so that it is 



\m + n 1 



Proceeding in this way Condorcet finally arrives at the conclu- 

 sion that the probability of having A is ^ and the probability of 



