Theory of Probabilities. 355 



Prob. A'B'E', = «'/3', 

 „ AB'E, = a$% 

 „ AB'E', = ^X', 

 „ AM, = «^, 

 n A'BE', = a '^, 

 „ ABE, = ol@{\+il—\il), 

 „ ABE', = apX/fL*. 

 It will be remembered that 



Prob. A'B'E = 0.. 



Prof. Boole's solution, which is inconsistent with the foregoing 

 one, is given in his well-known work, l An Investigation of the 

 Laws of Thought/ &c. Dublin, 1854, p. 321 et seq. Although 

 given as a solution of the causation statement of the question, as 

 already remarked, it seems to be (and I think Prof. Boole would 

 say that it is) a solution of the concomitance statement of the 

 question. It is certainly a most remarkable and suggestive one; 

 I am strongly inclined to believe that it is correct ; which of 

 course does not interfere with the correctness of my solution, if 

 the two really belong to distinct questions. 



I reproduce, Prof. Boole's solution, without attempting to ex- 

 plain (indeed I do not understand to my own satisfaction) the 

 logical principles upon which it is based. It is conducted by 

 means of the auxiliary quantities x, y, s, t, which are quantities 

 replacing logical symbols originally represented by the same let- 

 ters. I will designate these quantities, without attempting to 

 explain the meaning of the term, as Boolian Probabilities, viz. 



( Boolian Prob. A = x, 



AE = s, 



B = y, 



BE = t; 



and, as before, x\ &c. are used to denote 1— a?, &c. The 

 required probability of E is taken to be u. 



The event A can, by the data of the question, only happen 

 in concomitance as follows : viz. the concomitant events are 



Boolian Probs. 

 A. AE .B.BE xsyt 



A. AE.B'(BE)' xsy't* 



A.(AE)' . (BE)' xsU\ 



which may be analysed as follows : viz. if A and AE, then either 

 B and BE or B' and (BE)'; but if A and (AE)', then (B or B' 

 but of necessity) (BE)'. And this being so, the sum 

 xsyt + xsijt 1 + ocsH' 

 2B3 



