HUYGENS. 25 



throw, then B has two throAvs, then A two throws, and so on until 

 one or the other gains. Shew that ^'s chance is to J5's as 10355 to 

 12276. 



(2) Three players A, B, C take twelve balls, eight of which 

 are black and four white. They play on the following condition ; 

 they are to draw blindfold, and the first who draws a white ball 

 wins. A is to have the first turn, B the next, G the next, then 

 A again, and so on. Determine the chances of the players. 



Bernoulli solves this on three suppositions as to the meaning ; 

 first he supposes that each ball is replaced after it is drawn ; 

 secondly he supposes that there is only one set of twelve balls, 

 and that the balls are not replaced after being drawn ; thirdly he 

 supposes that each player has his own set of twelve balls, and that 

 the balls are not replaced after being drawn. 



(3) There are forty cards forming four sets each of ten cards ; 

 A plays with B and undertakes in drawing four cards to obtain 

 one of each set. Shew that ^'s chance is to -S's as 1000 is to 8139. 



(4) Twelve balls are taken, eight of which are black and four 

 are white. A pla3^s with B and undertakes in drawing seven balls 

 blindfold to obtain three white balls. Compare the chances of 

 A and B. 



(5) A and B take each twelve counters and play with three 

 dice on this condition, that if eleven is throAA-n A gives a counter 

 to B, and if fourteen is thrown B gives a counter to A ; and he 

 wins the game who first obtains all the counters. Shew that A 's 

 chance is to ^'s as 244140625 is to 282429536481. 



oQ>. The treatise by Huygens continued to form the best 

 account of the subject until it was superseded by the more elabo- 

 rate works of James Bernoulli, Montmort, and De Moivre. Before 

 we speak of these we shall give some account of the history of the 

 theory of combinations, and of the inquiries into the laws of 

 mortality and the principles of life insurance, and notices of 

 various miscellaneous investigations. 



