CONDOBCET. 383 



As is very common with Condorcet, it would be uncertain from 

 his language what questions he proposed to consider. On examin- 

 ing his solution it appears that his 1 and 3 are absolutely identical, 

 and that his 2 and 4 differ only in notation. 



703. In his sixth problem Condorcet says that he proposes the 

 same questions as in his fifth problem, taking now the hypothesis 

 that the probability is not constant. 



Here his 1 and 3 are really different, and his 2 and 4 are really 

 different. 



It seems to me that no value can be attributed to the discus- 

 sions which constitute the problems from the second to the sixth 

 inclusive of this part of Condorcet's work. See also Cournot's 

 Exposition de la Theorie cles Chances... -psige 166. 



704. The seventh problem is an extension of the first. Sup- 

 pose there are two events A and N, which are mutually exclusive, 

 and that in m + n trials A has happened m times, and N has hap- 

 pened n times : required the probability that in the next p +q 

 trials A will happen j; times and N happen q times. 



Suppose that x and 1 — x were the chances of A and JSf s.t a 

 single trial ; then the probability that in m + n trials A would 

 happen m times and iV^ happen n times would be proportional to 

 x"' (1 — xy. Hence, by the rule for estimating the probabilities of 

 causes from effects, the probability that the chance of A lies be- 

 tween X and x + dx at a single trial is 



x'^Q.-xydx 



{ r»'" (1 - xf dx y 



J 



And if the chance of ^ at a single trial is x the probability 

 that mp-\-q trials A will occur j) times and N occur q times is 



^===-x'{\-xy. 

 ^^ , . 



Hence finally the probability required in the problem is 



, , ^ \ x"^-"' il - xY^'' dx 

 \P±± K 



\e[i fx^'{i-xydx 



