CONDORCET. 861 



Jusqu'ici nous avons suppose iin seul Tribunal ; dans plusieurs pays 

 cependant on fait juger la meme affaire par plusieurs Tribunaux, ou 

 plusieurs fois par le meme, mais d'apres une nouvelle instruction, jus- 

 qu'^ ce qu'on ait obtenu un certain nombre de decisions conformes. 

 Cette bypotbese se subdivise en plusieurs cas differens que nous aliens 

 examiner separement. En effet, on peut exiger, 1". I'unanimite de ces 

 decisions ; 2°. une certaine loi de pluralite, formee ou par un nombre 

 absolu, ou par un nombre proportionnel au nombre des decisions 

 prises ; 3^ un certain nombre consecutif de decisions conformes. Quand 

 la forme des Tribunaux est telle, que la decision peut etre nulle, comma 

 dans la septieme hypotbese, il faut avoir ^gard aux decisions nulles. 

 Enfin il faut examiner ces differens cas, en supposant le nombre de ces 

 decisions successives, ou comme determine, ou comme indefini. 



677. The ninth Hypothesis extends over pages 57 — 86 ; it 

 appears to have been considered of gi'eat importance by Condorcet 

 himself We shall give some detail respecting one very in- 

 teresting case which is discussed. This case Condorcet gives on 

 pages 73 — 86. Condorcet is examining the probability of the 

 correctness of a decision which has been confirmed in succession 

 by an assigned number of tribunals out of a series to which the 

 question has been referred. The essential part of the discussion 

 consists in the solution of two problems which we will now enun- 

 ciate. Suppose that the probability of the happening of an event 

 in a single trial is v, and the probability of its failing is e, required, 

 1st the probability that in r trials the event will happen p times 

 in succession, 2nd the probability that in r trials the event will 

 happen p times in succession before it fails p times in succession. 



It is the second of these problems which Condorcet wishes 

 to apply, but he finds it convenient to begin with the solution 

 of the first, which is much the simpler, and which, as we have 

 seen, in Art. 325, had engaged the attention of De Moivre. 



678. We have already solved the first problem, in Art. 325, 

 but it will be convenient to give another solution. 



Let (/) {r) denote the probability that in r trials the event will 

 happen^ times in succession. Then we shall have 



^ (r) =ifJ^v^~' e<j)(r-p)+ if^e ^ (r -^ + 1) + ... 



,..+ve(j){r-2)+e(j>{r-l) (1). 



