608 PEOFESSOR BOOLE ON THE COMBINATION 



we have the following equations : — 



u + fX = q 1 (9) 



u+ v =r J 



with the inequations 



w>0, X>0, //>0, v>0, u + n + v + \ <1 . (10) 

 Determining from the equations X, /x, v, and substituting in the inequations, we get 



«>0 p — m>0, q — w>0, r — tt>0, p + ^ + r— 2m <1 . (11) 

 a system which agrees with that obtained by the previous investigation (5) Art. 13. 

 14. The general rule for the determination of the conditions of possible expe- 

 rience and of limitation in a question of probability may be thus stated. 



Resolve the events whose probabilities are either given or sought, into the 

 mutually exclusive alternatives which they involve. If the calculus of Logic is 

 employed, this is done by development. 



Represent the probabilities of these alternations by X, fx. v, &c, and express 

 the probabilities given and sought by the corresponding sums of these quantities. 

 This will furnish a series of equations, which we will suppose to be n in number. 

 Determine from these equations any n of the quantities X, fx, v, in terms of the 

 others. 



Substitute the values thus obtained in the inequations 



A>0 //>0 v>0 (1) 



K+fx + v ^1 (2) 



Eliminate in succession such of the quantities X, /jl, v, . . as are left in the above 

 inequations after the substitution. 



The elimination of any quantity as t from the inequations, is effected by re- 

 ducing each inequation to the form t < a, or to the form t > b, and observing that 

 two such forms as the above give a > b. 



If the " alternations" into which the events whose probabilities are given or 

 sought are resolved, extend to all possible combinations of the simple events out 

 of which they are formed, the inequations (2), must be replaced by the equation 



A + /X + V . . =1 (3) 



The rest of the process will be the same as before. 



In the form of the above method developed in the Philosophical Magazine the 

 quantities X, /x, v, . . represent the probabilities, not of those alternations alone, 

 which are contained in the events whose probabilities are given or sought, but of all 

 possible alternations which can be formed, by combining the simple events x,y,z.. 

 In this form, therefore, we have always an equation of the form (3), in the place 

 of an inequation of the form (2). But though the result is the same, the form 

 given to the method in this section is to be preferred, as it requires the elimina- 



