366 CONDOKCET. 



On his page 75 Condorcet gives an equivalent result without 

 explicitly using (7) ; but he affords very little help in establish- 

 ing it. 



Let X (^') tlenote what </> (r) becomes when v and e are inter- 

 changed ; that is let % (r) denote the probability that in r trials 

 the event will fail^ times in succession before it happens ^ times 

 in succession. 



Let E denote the value of % (r) when r is infinite. Then we 

 can deduce the value of E from that of V by interchanging v and 

 e ; and we shall have V+ E= 1, as we might anticipate from the 

 result at the end of Art. 679. 



Condorcet says that we shall have 



where f is une fonction semhlable de v et de e. 



Thus it would appear that he had some way of arriving at 

 these results less simple than that which we have employed ; for 

 in our way we assign V and E definitely. 



It will be seen that 



E ~ e-' l-v"' 



and this is less than — if v be greater than e. 



We have then two results, namely 



^_(£)_^ V tf 



the first of these results is obvious and the second has just been 

 demonstrated. From these two results Condorcet seems to draw 



the inference that , ; continually diminishes as r increases ; see 



his page 78. The statement thus made may be true but it is not 

 demonstrated. 



Condorcet says on his page 78, La probabilite en general que 

 la decision sera en faveur de la vcrite, sera exprimee par 



^^ (1 -v)[l- e") 

 e" (1 -e){l- v^) * 



