EULER. 94.7 



It will be found tliat tliis is 



(n - 2) (n - 3 ) {n - 4 ) 

 1.2^3 • 



The chances of the three events will be found by dividing 

 the number of ways in which they can respectively occur by the 

 whole number. 



Thus we obtain for I, ii, iii, respectively 



2-3 2.3(^-3) {n - 3) {n - 4) 



71 (a -1)' n{a-l) ' n (n ~ 1) ' 



441. Euler's next memoir also relates to a lottery. This 

 memoir is entitled Solution d'lme question tres difficile dans le 

 Calcid des Prohahilites. It w^as published in the volume for 

 1769 of the Histoire de VAcad. ... Berlin; the date of publication 

 is 1771 : the memoir occupies pages 285 — 302 of the volume. 



442. The first sentences give a notion of the nature of the 

 problem. 



C'est le plan d'une lotterie qni ni'a fourni cette question, que je 

 me propose de developper. Cette lotterie etoit de cinq classes, chacuue 

 de 10000 billets, parmi lesquels il y avoit 1000 prix dans chaque 

 classe, et par consequent 9000 bJancs. Chaque billet devoit passer 

 par toutes les cinq classes; et cette lotterie avoit cela de particulier 

 qu'outre les prix de chaque classe on s'engagooit de payer un ducat 

 a cliacun de ceux dont les billets auroient passe par toutes les cinq classes 

 sans rien gagner. 



443. We may put it perhaps more clearly thus. A man 

 takes the same ticket in 5 different lotteries, each having 1000 

 prizes to 9000 blanks. Besides his chance of the prizes, he is to 

 have £1 returned to him if he gains no prize. 



The question which Euler discusses is to determine the pro- 

 bable sum which will thus have to be paid to those who fail 

 in obtaining jmzes. 



444. Euler's solution is very ingenious. Suppose h the num- 

 ber of classes in the lottery ; let n be the number of prizes in each 

 class, and m the number of blanks. 



