424 Mr DE morgan, ON A QUESTION 



inferred by reasoning from the observed event to the probability of 

 its cause. That is, they assume in effect that the probability of the 

 equation (p {x, y) = a, where x is given and y presumed, must be the 

 same as in the case where y is given and x presumed. The preceding- 

 formula is neither admissible upon the reasoning produced, nor in fact 

 correct : as the following investigation will shew. 



There having been made n (or v ■{■ w) trials, at each of which 

 either A or JB must have happened; and A having happened v times, 

 and B w times: requu-ed the presumption that the probability of A 

 happening lay between two given limits a and b (b > a). 



The presumption that this probability lies between a and b, is 

 /„V(1 -xfdx 



fo'x'ii -xydx' ^^^ 



to the approximate determination of which, when v + w is a con- 

 siderable number, I proceed to apply the method of Laplace. 



Let y be a function of x which vanishes when x = and when 

 a; = 1 ; and let V, the intermediate maximum value of y, correspond 

 to X = X. Assume y — Fe"' , so that Avhile x increases from to X, 

 and from X to 1, t shall increase from — oc to 0, and from to + <x. 

 Let X = X + 9, and determine from 



^ + ^" ? + ^"' O + ^'"" sTTTi + = ^^" ' (^) 



V being = 0, since F is a maximum value of y. 



Let this process give 



9 = Bj + B,f + B,t' + 



and let x = a, and x = b, give t = fx, and t = v. 



Then, since clx = d9 

 f^ydx = V{Bj;;e-''dt + 2Bj;e-''fdf + SBJ^e-'' t^df + \ 



= F(JB, + I ^3 + ...)/;e-'V^ + V(B., + Y ^. + ...) 6-"" 



-F(^3 + |^J?3 + ...)6-''. (4) 



