developed in Professor Boole's " Laws of Thought." 473 



Of the eleven equations given by the independence of the simple 

 events, only two involve merely terms comprised in V, and con- 

 sequently those two arc the only necessary assumptions. The 

 truth of the remaining nine is immaterial to the question. The 

 two which affect the terms in V only are 



^=^,and:^ = l 



/U. ft) V w 



As the events represented by s and xz are concomitant, and also 

 those represented by t and yz, the event S is equivalent to Aj, 

 Aj and E all happening, fju to Ag and E not Aj, p to A, E not Aj, 

 T to Aj Ag not Ej V to Aj not Ag or E, ^ to Ag not A, or E, 

 (o to neither Aj Ag nor E. Consequently the two equations 

 assumed by Professor Boole in virtue of the method he employs 

 are 



Prob. of Aj, Agj and E all happening _ Prob. Aj, not Ag, E 



Prob. not Aj, A^, E Prob. not Aj, not Ag, not E' 



and 



Prob A,, Ag, not E _ Prob. Aj, not Ag, not E 

 Prob. not A,, Ag, not E ~ Prob. not A„ not Ag, not E' 



These two conditions being assumed, it is easy by common 

 algebra to determine the question ; for, besides the six equations 

 given, as I said before, in the data, we have the two 



7} a rf a' 



From the first five and these last two it is easy to eliminate 

 ^> V) v'> ^> ^'} ^^^ <^'> leaving a quadratic in f ; and in this the 

 value CiT^i + Cg/Jg""^ must be substituted for ^, giving a qua- 

 dratic to determine u similar to that found by Professor Boole. 



The second of these tw'O assumed equations, though perfectly 

 arbitrary, is perhaps not an unreasonable one. It asserts that 

 in those cases in which E does not occur, the relation of inde- 

 pendence subsists between Aj and Agj that is, that provided E 

 do not occur, Aj is as likely to happen if Ag happen as if Ag fail. 

 I do not see, however, that it is a more reasonable or probable 

 hypothesis than others that might be framed ; for instance, than 

 those assumed by Mr. Cayley in his memoir in this Magazine. 

 But the first of these equations appears to me not only arbitrary 

 but eminently anomalous. In the form in which it stands as a 

 relation amoug the chances of A,, Ag and E, no one, I should 

 think, can contend that it is either deduced from the data of the 

 problem, or that the mind by the operation of any law of thought 

 recognizes it as a necessary or most reasonable assumption. 

 Neither can it be said that the mutual independence of the events 



