32 Solution of a Question in the Theory of Probabilities. 



and not greater than any one of the three quantities 



l-«(l-7>), 1-^(1 -ry), up + Pq. . . (7) 



Moreover, whenever the data of the problem are real, i. e. repre- 

 sent a possible experience, there will exist one root, and only 

 one root, of the above equation satisfying the conditions described. 

 The conditions of a possible experience are, that each of the 

 three quantities in (7) exceeds each of the two in (6). 



The above solution may readily be verified in the case con- 

 sidered by Mr. Cayley, and in the three cases discussed in this 

 paper. Of these I will here confine myself to the last, 



Clearing the equation of fractions, and making jo = l, q = 0, 

 we have 



{l—'u){l-l3-u){u-u) = (l-u){u-u)u. 



This equation is satisfied by the values 



but the conditions above stated restrict our choice to the first, 

 which we have before seen to be the true one. 



3rd. Upon the nature of the errors which are most to be 

 apprehended in the solution of questions in the theory of pro- 

 babilities, I will only remark that they are not usually mathe- 

 matical, in the ordinary sense of that term, but arise fx'om the 

 necessity of employing a logic of a peculiarly subtle or highly 

 complex character. When the data are the probabilities of in- 

 dependent simple events, the method of procedure is sufliciently 

 easy ; but if those data relate to events occurring in combina- 

 tions, or connected by causal relations, the principles which 

 suffice for the former case become either inadequate or inappli- 

 cable. Laplace has to some extent investigated the additional 

 new principles (derivable from the prior definitions and axioms 

 of the science) of which it is then necessary to take account. 

 But all these aids carry us but a short way in advance ; and of 

 this I am fully assured, that no general method for the solution 

 of questions in the theory of probabilities can be established 

 which does not explicitly recognize, not only the special nume- 

 rical bases of the science, but also those universal laws of thought 

 which are the basis of all reasoning, and which, whatever they 

 may be as to their essence, are at least mathematical as to their 

 form. Such a method I have exhibited in a treatise now on the 

 eve of publication, and to which I must refer for the investiga- 

 tion of the problem, the solution of which has been exemplified 

 in this paper. 



5 Grenville Place, Cork, 

 Nov. 30, 1853. 



