on the Theory of Chances. 89 



kind of statement, progression and results. At present I merely 

 ojQfer these observations as preliminary to the question which I 

 am called upon now to consider. 



Mr. Wilbraham's remarks chiefly apply to two solutions of the 

 following problem respectively published in this Journal by 

 Mr. Cayley and myself. " The probabilities of two events Aj, 

 Ag are Cj and c^ respectively. The probability that if A, present 

 itself another event E will accompany it i^p^, and the probabiUty 

 that if Ag present itself E will accompany it is p^. The event E 

 cannot happen in the absence of Aj and A^, but of the connexion 

 of the latter events nothing is known. Required the probability 

 of E.^^ (See also Laws of Thought, p. 321.) Representing 

 Aj, Ag, and E by x, y, z respectively, the data of this problem are 



Prob . x—c^ Prob . y^c^ Prob . xz^c^p^ Prob . yz = Cc^p,^ 



Prob. 5'(l-^)(l-2/) = 0. 



Mr. Wilbraham shows that both Mr. Cayley's solution and my 

 own introduce two equations. To this I remark in passing, 

 that there can be no objection so long as the equations in ques- 

 tion are consequences of the laws of thought and expectation as 

 applied to the actual data. Respecting the equations involved 

 in my own solution, Mr. Wilbraham remarks : — " The second of 

 these two assumed equations, though perfectly arbitrary, is per- 

 haps not an unreasonable one I do not, however, see that 



it is a more reasonable or probable hypothesis than others that 

 might be framed; for instance, than those assumed by Mr. Cay- 

 ley in his memoir in this Magazine. But the first of these equa- 

 tions appears to me not only arbitrary but eminently anoma- 

 lous.^' After this he deduces the equations which represent in 

 a similar manner Mr. Cayley's hypotheses. 



I should be reluctant to enter into any comparison of Mr.^ 

 Cayley's solution and my own if the above remarks did not 

 render it necessary to the interests of truth. It cannot be doubted 

 that Mr. Cayley's solution is erroneous. Granting for a moment 

 that both solutions involve hypotheses, there is this diiFerence 

 between, them (a difference passed over in silence by Mr. Wil- 

 braham), that Mr. Cayley's hypotheses lead to results absolutely 

 inconsistent with the data — that my own hypotheses do not. 

 One case easily tested is when we havejOj = l, and at the same 

 timej02=0. Another and more general case is when the con- 

 stants are so related that we have either 



CiPi + Cc,{l-p^) = \, 

 or 



I would refer on these points to a paper " On the conditions by 



