300 BAYES. 



Art. 551, there has however often been no adequate ground for 

 such knowledge or assumption. 



553. We have already stated that Bayes gave two rules for 

 approximating to the value of the area which corresponds to the 

 integral. In the first memoir, Price suppressed the demonstrations 

 to save room ; in the second memoir, Bayes's demonstration of the 

 principal rule is given : Price himself also continues the subject. 

 These investigations are very laborious, especially Price's. 



The following are among the most definite results which Price 

 gives. Let n =p + q, and suppose that neither p nor q is small ; 



let h = — //IN • Then if an event has happened p times and 



failed q times, the odds are about 1 to 1 that its chance at 



a single trial lies between - + -7^ and 7^ ; the odds are about 



2 to 1 that its chance at a single trial lies between - ■\- h and 



n 



-^ — h'. the odds are about 5 to 1 that its chance at a sinsfle 

 n ° 



trial lies between ^ + A V2 and ^ — h J% These results may be 



n n "^ 



verified by Laplace's method of approximating to the value of the 



definite integrals on which they depend. 



554. We may observe that the curve y — x^ (1— xy has two 

 points of inflexion, the ordinates of which are equidistant from the 

 maximum ordinate ; the distance is equal to the quantity h of the 

 preceding Article. These points of inflexion are of importance in 

 the methods of Bayes and Price. 



