60 JAMES BERNOULLI. 



103. The tenth problem is to find in how many trials one 

 may undertake to throw a six with a common die. James Bernoulli 

 gives a note in reply to an objection which he suggests might 

 be urged against the result; the reply is perhaps only intended 

 as a popular illustration : it has been criticized by Prevost in the 

 NoiLveaux Memoir es de FA cad.... Berlin for 1781. 



104. James Bernoulli gives the general expression for the 



chance of succeeding m times at least in n trials, when the chance 



of success in a single trial is known. Let the chances of success 



b c 



and failure in a single trial be - and - respectively: then the 



required chance consists of the terms of the expansion of - + — ) 



from ( - j to the term which involves - j [ - J , both inclusive. 



This formula involves a solution of the Problem of Points for 

 two players of unequal skill; but James Bernoulli does not ex- 

 plicitly make the application. 



105. James Bernoulli solves four of the five problems which 

 Huygens had placed at the end of his treatise ; the solution of the 

 fourth problem he postpones to the third part of his book as it 

 depends on combinations. 



106. Perhaps however the most valuable contribution to the 

 subject which this part of the work contains is a method of solving 

 problems in chances which James Bernoulli speaks of as his own, 

 and which he frequently uses. We will give his solution of the 

 problem which forms the fourteenth proposition of the treatise 

 of Huygens : we have already given the solution of Huygens him- 

 self; see Art. 34. 



Instead of two players conceive an infinite number of players 

 each of whom is to have one throw in turn. The game is to 

 end as soon as a player whose turn is denoted by an odd number 

 throws a six, or a player whose turn is denoted by an even number 

 throws a seven, and such player is to receive the whole sum at 

 stake. Let h denote the number of ways in which six can be 

 thrown, c the number of ways in which six can fail; so that 6 = 5, 



