NUMERICALLY DEFINITE REASONING. 349 



Wilbraham'^, in his criticism on Boole^s ^Method of Pro- 

 babilities/ '^that the really determinate problems solved 

 in the book^, as 2 and 3 of chapter xviii.^ might be more 

 shortly solved/" Boole remarks_, indeed, that they do not 

 fall directly within the scope of known methods ; but I 

 conceive that my logical symbols and method furnish all 

 that is required. 



27. In a similar manner we may solve the second of 

 Boole^s examples referred to by Mr. Wilbraham ; this is 

 as follows : — 



" The probability that one or both of two events happen 

 is j», that one or both of them fail is q. What is the pro- 

 bability that only one of these happens ? " 



Using a, /3, y, S to denote the probabilities of the four 

 obvious conjunctions of events, as before, we have the 

 data, 



a + /5 + r=i?, 



a + /3 + y + S=i. 

 The probability required is ff + y, and 



This is Mr. Boole^s result, obtained by him in a much 

 more complicated manner. 



28. This simple substitution of the probability of an 

 event for its logical symbol cannot be valid, however, if 

 there be any connexion between the events which renders 

 one more or less likely to happen when the other happens. 

 The probabilities of A and B being p and g, the proba- 

 bility of AB ispq, under the supposition that B is just as 

 likely to happen when A happens as when A does not 

 happen, and similarly that A is just as likely to happen 



* Philosophical Magazine, 4th Series, vol. vii. p. 465 ; vol. viii. p. 91. 



