

THE 

 LONDON, EDINBURGH and DUBLIN 



PHILOSOPHICAL MAGAZINE 



JOURNAL OF SCIENCE. 



SUPPLEMENT to VOL. VII. FOURTH SERIES. 





LXXI V. On the Theory of Chances developed in Professor Boole's 

 " Laws of Thought." By Henry Wilbraham, M.A., Felloiv 

 of Trinity College, Cambridge^. 



SOME communications having already within the last few 

 months appeared in the Philosophical Magazine, by Mr. 

 Cayley and Professor Boole, relating to the subjects treated in 

 the work lately published by the latter on the Laws of Thought, 

 it may be considered not out of place to publish in the pages of 

 the same Magazine the following observations on the theory of 

 chances developed in that work. 



The object of this paper is to show that Professor Boole does iu 

 the greater number of questions relating to chances solvable by his 

 method (or at least in those which are most difficult to treat by 

 other methods), tacitly assume certain conditions expressible by 

 algebraical equations, over and above the conditions expressed by 

 the data of the problem, and to show how these assumed con- 

 ditions may be algebraically expressed. 



When no condition among the chances of the simple events, 



I but only the absolute chances of tliose several simple events 

 J^l are given, the reasoning of Chapter XVIL of Prof. Boole's 

 '8 book shows that it is assumed that the events are independent, 

 i. e. that the event A is as likely to happen in one state of cir- 

 cumstances as regards the remaining events as in another ; for 

 . , instance, that A is as likely to happen if B happen and C do not, 

 1^ i as it is if B and C both happen or both fail; and this assump- 

 — ' tion is implicitly introduced in the logical method of working 

 - I the problem. It is an assumption easily expressed by an alge- 

 braical equation or system of equations. For instance, take the 

 most simple case, — there are two events A and B, the chance of 

 \ happening is a, of B, Z/ : what is the chance of A and B both 

 happening? There arc four possible cases; viz. (1) both liap- 

 ^^__^ jpcning, (2) B happening without A, (3) A without B, (4) both 



* Communicated by the Author, 

 Phil. Mag. S. 4. No. 48. Suppl. Vol. 7. 2 1 



