408 CONDORCET. 



will be found to be n — m + 1 if r is less than m + l, and if r is 

 greater than n + 1, and in other cases to be w - r + 1. 



748, Hence the probability that the duration of seven reigns 

 will amount to just 257 years is the coefficient of ic^" in the expan- 

 sion of the seventh power of 



{n - m + 1) a^ (1 - x) -^^ x"""^ 

 (1 — x) — ^ — (n - m + 1) 



Now Condorcet takes n = 60 and m = SO; and he says that the 

 value of the required coefficient is '000792, which we will assume 

 he has calculated correctly. 



Thus he has obtained the probability in the ordinary sense, 

 which he denotes by P; he requires the j)rohahilite projyre. He 

 considers there are 414 events possible, as the reigns may have 

 any duration in years between 7 and 420. Thus the mean proba- 



.1 -P 



bility of all the other events is , ; and so the prohahilite p^opre 



413P , 1 



1 ^ 412P ' ^i^^^^^^^S- 



IS 



749. Condorcet says that other historians assign 140 years in- 

 stead of 257 years for the duration of the reigns of the kings. 

 He says the ordinary probability of this is '008887, which we 

 may denote by Q. He then makes the prohahilite propre to be 



which IS more than - . 



He seems here to take 413, and not 414, as the whole number 

 of events. 



750. Condorcet then proceeds to compare three events, namely 

 that of 257 years' duration, that of 140 years' duration, and what 

 he calls wi autre dvenement mdetermine quelconque qui auroit pu 

 avoir lieu. He makes the prohahilites jyrojn-es to be respectively 



411P 411^ 1-P-Q 



410 (P+ Q) + i' 410 (p+ Q) + 1 ""'"'^ n^pToy+i' 



3 37 10 



which are approximately — , , '-— , -— . 



•^ 50 oO 50 



