CHAP. XVII.] GENERAL METHOD IN PROBABILITIES. 261 



prospective, and will be made manifest in a subsequent propo- 

 sition. 



PROPOSITION II. 



10. It is knovm that the probabilities of certain simple events 

 x, y, z 9 . . are p, q,r, . . respectively when a certain condition F is 

 satisfied; V being in expression a function ofx,y 9 z,.. Required 

 the absolute probabilities of the events x 9 y 9 z 9 . . 9 that is, the 

 probabilities of their respective occurrence independently of the con- 

 dition V. 



Let p 9 </, r, &c., be the probabilities required, i. e. the pro- 

 babilities of the events x, y, z, . . , regarded not only as simple, 

 but as independent events. Then by Prop. i. the probabilities 

 that these events will occur when the condition F, represented 

 by the logical equation V-\ 9 is satisfied, are 



\xV} \yV} [zF] 



TFT- IF!' IT]' 



in which [x F] denotes the result obtained by multiplying F by 

 x, according to the rules of the Calculus of Logic, and changing 

 in the result x, y, z 9 into p, q', r' 9 &c. But the above condi- 

 tioned probabilities are by hypothesis equal to p, q, r, . . re- 

 spectively. Hence we have, 



|>F] [yF] [zF] 



' ' 



from which system of equations equal in number to the quanti- 

 ties p' 9 q' 9 r', . . , the values of those quantities may be deter- 

 mined. 



Now x V consists simply of those constituents in F of which 

 a; is a factor. Let this sum be represented by V X9 and in like 

 manner let y V be represented by V y9 &c. Our equations then 

 assume the form 



, .,. , 



where [ FJ denotes the results obtained by changing in V x the 

 symbols x, y 9 z 9 &c., into p, q 9 r 9 &c. 



To render the meaning of the general problem and the 



