MALLET. 339 



n persons, charging a certain sum for each ticket. He retains for 

 himself a portion of the money which he thus receives, say a ; the 

 remainder he distributes into n prizes which will be gained by 

 those who bought the tickets. He also offers a further inducement 

 to secure buyers of his tickets, for he engages to return a sum, say 

 5, to every ticket-holder who does not gain a prize. The jDrizes are 

 distributed in the following manner. In a box are placed n coun- 

 ters numbered respectively from 1 to n. A counter is drawn, and 

 a prize assigned to the ticket-holder whose number corresponds to 

 the number of the counter. The counter is then replaced in the box. 

 Another drawing is made and a prize assigned to the corresponding 

 ticket-holder. The counter is then replaced in the box. This pro- 

 cess is carried on until n drawings have been made ; and the prizes 

 are then exhausted. 



Hence, owing to the peculiar mode of drawing the lottery, one 

 person might gain more than one prize, or even gain them all ; for 

 the counter which bears his number might be drawn any number 

 of times, or even every time. 



The problem proposed is to find the advantage or disadvantage 

 of the director of the lottery. 



633. Montmort solved the problem in the following manner. 

 Consider one of the ticket-holders. The chance that this per- 

 son's number is never drawn throughout the whole process is 



) . If it is not drawn he is to receive h from the director ; 



so that his corresponding expectation is h ( J • ^ similar ex- 

 pectation exists for each of the ticket-holders, and the sum of these 

 expectations is the amount by which the director's gain is di- 

 minished. Thus the director's advantage is 



a — nh I j . 



In the case which Montmort notices h was equal to a, and n 

 was 20000 ; thus the director's advantage was negative, that is, it 

 was really a disadvantage. Before Montmort made a complete 

 investigation he saw that the director's position was bad, and he 



22—2 



