440 Prof. Boole on a Genei^al Method in 



problem relating to events may be represented by a correspond- 

 ing problem relating to the issuing of balls from an urn. If, in 

 such imagined problem, any events s, t, &c. so enter as that 

 nothing is known or can be inferred respecting their connexion, 

 they must be treated (Principle I.) as if they were independent, 

 and therefore the balls by whose issue they are represented must 

 be regarded as free from any nexus affecting their issue. On 

 the other hand, if the events s, t, &c. are subject to any condition 

 as D=0, such condition must be introduced by the supposition 

 of a neanis simply forbidding the issue of balls of the species D, 

 without affecting the freedom of the other balls. Such a nexus 

 we may suppose to be established by the attachment of every ball 

 of the species D by a thread to the walls of the urn. All pos- 

 sible issues are thus restricted to balls of the species A or B, so 

 that the condition D=0 is equivalent, as we have before seen, to 

 the condition A + B = 1 . 



The general problem may therefore be represented as follows : — 

 An urn contains balls whose species are expressed by means 

 of the qualities s, t, &c. and their opposites, concerning the 

 connexion of which qualities nothing is known. Suddenly all 

 balls of the species D are attached by threads to the walls of the 

 urn, and this being done, there is a probability p that any ball 

 drawn is of the species s, a probability q that it is of the species 

 t, and so on. What is the probability that it is of the species A, 

 supposing that A and D denote mutually exclusive species of 

 balls, each defined by means of the properties s, t and their 

 opposites ? 



Let us, for simplicity, represent A + B by V, and let us repre- 

 sent by Vg the aggregate of constituents in V of which s is a 

 factor, by V^ the aggregate of constituents of which ^ is a factor, 

 and so on. Then, according to the principles of the Calculus of 

 Logic, we shall have the following interpretations, viz. — 



V= that event which consists in the drawing of a ball which 



is not of the species D. 

 V,= that event which consists in the di'awing of a ball which 



is of the species s and is not of the species D. 

 V<= that event which consists in the drawing of a ball which 



is of the species t and is not of the species D. 



Now let the total number of balls in the urn be N, and let S be 



the number which are of the species s, T the number which are 



of the species /, &c. Hitherto s and / have been used only as 



logical symbols expressing events. Let us now introduce a new 



set of symbols s, t, &c., to be used in a quantitative acceptation, 



ST 

 to denote the numerical ratios :j^, ^j^, &c. Then we have 



