262 GENERAL METHOD IN PROBABILITIES. [CHAP. XVII. 



ciple of its solution more evident, let us take the following ex- 

 ample. Suppose that in the drawing of balls from an urn 

 attention had only been paid to those cases in which the balls 

 drawn were either of a particular colour, "white," or of a par- 

 ticular composition, " marble," or were marked by both these 

 characters, no record having been kept of those cases in which a 

 ball that was neither white nor of marble had been drawn. Let 

 it then have been found, that whenever the supposed condition 

 was satisfied, there was a probability/? that a white ball would be 

 drawn, and a probability q that a marble ball would be drawn : and 

 from these data alone let it be required to find the probability 

 that in the next drawing, without reference at all to the condi- 

 tion above mentioned, a white ball will be drawn ; also the pro- 

 bability that a marble ball will be drawn. 



Here if x represent the drawing of a white ball, y that of a 

 marble ball, the condition V will be represented by the logical 



function 



xy + x(l-y) + (1 -x)y. 

 Hence we have 



V x = xy + x(\-y) = x 9 V y = xy + (1 - x) y =y; 

 whence 



and the final equations of the problem are 



from which we find 



p+ q -1 , p +q -1 



' = - - - - , q '=- - - . 

 q p 



It is seen that p and q are respectively proportional to p and 

 q 9 as by Professor Donkin's principle they ought to be. The 

 solution of this class of problems might indeed, by a direct appli- 

 cation of that principle, be obtained. 



To meet a possible objection, I here remark, that the above 

 reasoning does not require that the drawings of a white and a 

 marble ball should be independent, in virtue of the physical con- 

 stitution of the balls. The assumption of their independence is 

 indeed involved in the solution, but it does not rest upon any 



