Questions in the Theory of Probabilities are limited, 95 

 And hence 



^2(1 -P^) > ^1 (1 -Pi) + OciO- -Pi) - 1 + w; ; 



or 



w 



w 



= l-Ci(l-i?i)J 



From the conditions (5) and (6), we spe that w has for its lower 

 limits, the expressions 



^1/^1 and C2P2, (7) 



and for its upper limits, the expressions 



CiPct + c^P2> 1—Ci(l—J0j) and 1 — ^2(1— jOg). . . (8) 



These are the conditions assigned in my treatise on the Laws 

 of Thought, p. 325. They show, that if it is our object to deter- 

 mine Prob. z or w, the solution, to he a correct one, must lead 

 us to a value of that quantity which shall exceed each of the 

 values assigned in (7), and fall short of each of those assigned 

 in (8). They show also that the data of the problem will only 

 represent a possible experience when each of the values in (7) 

 shall fall short of, or not exceed each of those in (8). 



There is a class of problems characterized by the circumstance 

 that the quantities X, //., v . . are fewer in number than the equa- 

 tions in which they enter, which treated by this method lead to 

 equations as well as inequations connecting the data with each 

 other and with the probability sought. Whenever, too, the pro- 

 bability sought can be expressed as a linear function of the pro- 

 babilities which are given, its actual expression will be deter- 

 mined by the above method, and it will agree with the result 

 which would be assigned by the general method in probabilities 

 (Laws of Thought, Chap. XVIL). To exemplify this, let us take 

 the following problem (Ibid. p. 279). 



Given Prob. a?=jo, Prob. y=q, Prob.(a?(l--y) +2^(1 — a?)) =:r, 

 to find the limits of Prob. ocy or w. 



Assume 



Prob. a??/=\ Prob. a?(l— 2/)=/Lt Prob. (l~-a7)y=:v 

 Prob. (l-a?)(l-2/)=p. 

 Then we liave as the conditions furnished by the data, 

 \+fi=p 

 X + v=q 



^ fj, + v:=r j>. (9) 



\ = w 



