298 BATES. 



have for the probability, before the first ball is thrown, that the 

 second event will happen^ times and fail ^ times 



a 



P <1 



C©'('-3' 



Cv«X/» 



549. We now arrive at the most important point of the essay. 

 Suppose we only know that the second event has happened p times 

 and failed q times, and that we wish to infer from this fact the 

 probable position of the line ^i^" which is to us unknown. The 

 probability that the distance of EF from AB lies between h 

 and c is 



I x'^ {a — xy dx 

 •h 



ra 



x^(a-xydx 



J 



This depends on Bayes's Proposition 5, which we have given 

 in our Art. 544. For let z denote the required probability ; 

 then 



z X probability of second event = probability of compound event. 



The probability of the compound event is given in Art. 547, 

 and the probability of the second event in Art. 548 j hence the 

 value of z follows. 



550. Bayes then proceeds to find the area of a certain curve, 

 or as we should say to integrate a certain expression. We have 



/ 



^i>+i ^ ^P^ q{q-l) x'^' 



X I jL ^ X) CLX —~ — — ^ ~I~ 



p + 1 1 J9+ 2 ' 1.2 ^ + 3 "• 



This series may be put in another form ; let u stand for 1 — x, 

 then the series is equivalent to 



jj + l'^p+l p + 2 '^ (p + 1) {p-^2) p + S 



q(q-l)(q-2) x^^r' 



{p + l){p-\-'2){p + '6) iJ + 4 ■^••• 



This may be verified by putting for u its value and rearranging 

 according to powers of x. Or if we differentiate the series with 



