PREYOST AND LHUILIER. 461 



3,4.5.6 1 



p= 2, q=S, r =4 ; the above result becomes ^ o^~n a > that is — . 



If we knew a priori that the die had as many faces ace as not-ace 

 we should have -^^ , that is ^5 ' ^^^' ^^^^ required chance. The para- 



dox is that q- is o-reater than — ; ; while the fact that we have had 

 14 lb 



only two aces out of five throws suggests that we ought to have a 



smaller chance for obtaining four consecutive aces, than we should 



have if we knew that the die had the same number of faces ace as 



not-ace. We need not give the explanation of the paradox, as it 



will be found in connexion with a similar example in Laplace, 



Theorie...des Proh. page cvi. 



857. The fourth section gives some mathematical develop- 

 ments. The following is the substance. Suppose n dice, each 

 having r faces ; and let the number of faces which are marked ace 

 be m, m\ m"\ . . . respectively. If a die is taken at random, the 

 probability of throwing ace is 



on -\-m -^-m + ... 



nr 

 If an ace has been thrown the probability of throwing ace again 

 on a second trial with the same die is 



m^ + m" + m"" + . . . 



r [in -{■ m -\- m + ...) 

 The first probability is the greater; for 



{m + m 4- m" + ...y is greater than n [m^ + m'"^ + m"'^ + ...). 

 The memoir demonstrates this simple inequality. 



858. Prevost and Lhuilier are also the authors of a memoir 

 entitled Me mo ire sur ^application du Calcid des prohahilites a la 

 valeur du temoignage. 



This memoir is published in the volume for 1797 of the Me- 

 moires de V Acad.... Berlin; the date of publication is 1800: the 

 memoir occupies pages 120 — 151 of the portion of the volume 

 devoted to speculative philosophy. 



The memoir begins thus : 



Le but de ce memoire est plutot de reconnoitre I'etat actuel de cette 

 theorie, que d'y rien aj outer de nouveau. 



