268 GENERAL METHOD IN PROBABILITIES. [CHAP. XVII. 



We may hence determine the nature of that new experience 

 from which the actual value of c may be obtained. For if we 

 substitute in C for , t, &c., their original expressions as func- 

 tions of the simple events x, y, z, &c., we shall form the ex- 

 pression of that event whose probability constitutes the denomi- 

 nator of the above value of c ; and if we multiply that expression 

 by the original expression of w, we shall form the expression of 

 that event whose probability constitutes the numerator of c, and 

 the ratio of the frequency of this event to that of the former one, de- 

 termined by new observations, will give the value of c. Let it be 

 remarked here, that the constant c does not necessarily make its 

 appearance in the solution of a problem. It is only when the 

 data are insufficient to render determinate the probability sought, 

 that this arbitrary element presents itself, and in this case it is 

 seen that the final logical equation (2) or (5) informs us how it 

 is to be determined. 



If that new experience by which c may be determined can- 

 not be obtained, we can still, by assigning to c its limiting values 

 and 1, determine the limits of the probability of w. These 

 are 



Minor limit of Prob. w = -== . 



a i M A + C 



Superior limit = ^ . 



Between these limits, it is certain that the probability sought 

 must lie independently of all new experience which does not ab- 

 solutely contradict the past. 



If the expression of the event C consists of many constituents, 

 the logical value of w being of the form 



^ ^ 



w = A + -C l + -C 2 + &c. 9 



we can, instead of employing their aggregate as above, present 

 the final solution in the form 



-r, , A + C^Ci + C 2 C 2 + &C. 



Jrrob. w = =^ _ 



