258 GENERAL METHOD IN PROBABILITIES. [CHAP. XVII. 



things denoted by the Proposition, " It either rains or thunders, 

 but not both ;" the expression of that state of things being 



If for brevity we represent the function (x, y, z 9 . .), used in 

 the above acceptation by F, it is evident (VI. 13) that the law 

 of duality 



F(l - F) = 0, 



will be identically satisfied. 



The simple events x, y, z will be said to be " conditioned" 

 when they are not free to occur in every possible combination ; 

 in other words, when some compound event depending upon 

 them is precluded from occurring. Thus the events denoted by 

 the propositions, " It rains," " It thunders," are "conditioned" 

 if the event denoted by the proposition, " It thunders, but does 

 not rain," is excluded from happening, so that the range of pos- 

 sible is less than the range of conceivable combination. Simple 

 unconditioned events are by definition independent. 



Any compound event is similarly said to be conditioned if it 

 is assumed that it can only occur under a certain condition, that 

 is, in combination with some other event constituting, by its pre- 

 sence, that condition. 



7. We shall proceed in the natural order of thought, from 

 simple and unconditioned, to compound and conditioned events. 



PROPOSITION I. 



1st. If p 9 q, r are the respective probabilities of any uncon- 

 ditioned simple events x, y, z, the probability of any compound 

 event F will be [F], this function [F] being formed by changing, 

 in the function F, the symbols x, y, z into p, q, r, fyc. 



2ndly. Under the same circumstances , the probability that if 

 the event V occur, any other event V will also occur, will be 



[W'l 



L -, J , wherein [FF 7 ] denotes the result obtained by multiplying 



together the logical functions V and V, and changing in the result 

 x, y, z, Sfc. into p, q, r, -c. 



Let us confine our attention in the first place to the pos- 



