30 Mr. G. Boole's Solution of a Question in 



matical Journal in the montli of November 1851. This may 

 justify my offering a few remai'ks upon the subject. 



The problem as considered by Mr. Cayley is thus stated. 



Given the probability « that a cause A will act, and the pro- 

 bability p that A acting the effect will happen ; also the proba- 

 bility /3 that a cause B will act, and the probability q that B 

 acting the effect will happen : required the total probabiUty of 

 the effect, supposed impossible in the absence of both the causes. 

 The solution given by Mr. Cayley is as follows. Let \ be the 

 probability that the cause A acting will act efficaciously, /a. the 

 probability that the cause B acting will act efficaciously. Then 



p=\+{l-\)f^l3 . (1) 



g = fM+{l-lx)\c^, . ■ (2) 



which determines X,, /a, and the total probability « of the effect 

 will be given by 



M=Xa-t-//,/3— X/iayS (3) 



Mr. Cayley shows that this leads to a correct result when 

 a=l. He fm-ther remarks, that the problem presents no dif- 

 ficulty. 



I think it to be one of the peculiar difficulties of the theory of 

 probabilities, that its difficulties sometimes are not seen. The 

 solution of a problem may appear to be conducted according to 

 the principles of the theory as usually stated ; it may lead to a 

 result susceptible of verification in particular instances ; .and yet 

 it may be an erroneous solution. The problem which ]\Ir. Cay- 

 ley has considered seems to me to afford a good illustration of 

 this remark. Several attempts at its solution have been for- 

 warded to me, all of them by mathematician^ of great eminence, 

 all of them admitting of particular verification, yet differing from 

 each other and from the truth. Mr. Cay ley's solution is the 

 only published one I have seen, and I feel I must extend to it 

 the same observations. But in doing this, I willingly add that 

 I have two or three times attempted to solve the problem by the 

 same kind of reasoning, and have not approached so near the 

 truth as Mr. Cayley has done. To illustrate these remarks, I 

 will first complete Mr. Cayley's solution, and give one or two 

 apparent verifications, then exhibit the true solution j and lastly, 

 make a few observations upon the general subject. 



1st. To complete Mr. Cayley's solution, eliminate \ and /u, 

 from (1), (2), and (3) ; the result is 



a quadratic equation from which u must be determined. Sup- 

 pose that j9=i and q=l, the above equation gives 



