24 ^ HUYGENS. 



84. The fourteenth proposition consists of the following 

 problem. A and B play with two dice on the condition that A 

 is to have the stake if he throws six before B throws seven, and 

 that B is to have the stake if he throws seven before A throws 

 six ; ^ is to begin, and they are to throw alternately ; compare 

 the chances of A and B. 



We will give the solution of Huygens. Let B's chance be 

 worth X, and the stake a, so that a — a? is the worth of ^'s chance ; 

 then whenever it is ^.'s turn to throw x will express the value 

 of B's chance, but when it is i>'s own turn to throw his chance 

 will have a different value, say ?/. Suppose then A is about to 

 throw ; there are 36 equally likely cases ; in 5 cases A wins and B 

 takes nothing, in the other 81 cases A loses and B's turn comes 

 on, which is worth y by supposition. So that by the third propo- 

 sition of the treatise the expectation of B is ^ — - , that is, 



^2l, Thus 

 So 81?/ 



Now suppose B about to throw, and let us estimate ^'s chance. 

 There are S6 equally likely cases ; in 6 cases B wins and A takes 

 nothing ; in the other 80 cases B loses and ^'s turn comes on 

 again, in which case B's chance is worth x by supposition. So 



that the expectation of B is — ^j^ — . Thus 



(ja-^SOx 



81« 



From these equations it will be found that x = -^ , and thus 



80cj 



a — x=^ 



61 



, so that ^'s chance is to ^'s chance as 80 is to 81. 



85. At the end of his treatise Huygens gives five problems 

 without analysis or demonstration, which he leaves to the reader. 

 Solutions are given by Bernoulli in the Ars Conjectandi. The 

 following are the problems. 



(1) A and B play with two dice on this condition, that A gains 

 if he throws six, and B gains if he throws seven. A first has one 



