Theory of Probabilities. 361 



respectively. But this is an immediate consequence of the given 

 values prob. A = a, prob. B=/3, and of the deduced equation 

 prob. AB=0. 



2 Stone Buildings, W.C., 

 March 18, 1862. 



The foregoing paper was submitted to Prof. Boole, who, in a 

 letter dated March 26, 1862, writes : — 



" The observations which have occurred to me after studying 

 your paper are the following. 



1st. " I think that your solution is correct under conditions 

 partly expressed and partly implied. The one to which you 

 direct attention is the assumed independence of the causes de- 

 noted by A and B. Now I am not sure that I can state pre- 

 cisely what the others are ; but one at least appears to me to be 

 the assumed independence of the events of which the probabili- 

 ties according to your hypothesis are oCk, /3/z. Assuming the 

 independence of the causes as to happening, I do not think that 

 you are entitled on that ground to assume their independence as 

 to acting \ because, to confine our observations to common expe- 

 rience, we often notice that states of things apparently indepen- 

 dent as to their occurrence, may, when concurring, aid or hinder 

 each other in such a manner that the one may be more or less 

 likely to act ' efficiently ' in the presence of the other than in 

 its absence. I use the language of your own hypothesis of effi- 

 cient action. 



" 2ndly. When I say that I think your solution correct under 

 certain conditions, I ought to add that it appears to me that 

 such conditions ought to be stated as part of the original data, 

 and that they ought to be of such a kind that they can be esta- 

 blished by experience in the same way as the other data are. 

 For instance, if experience, as embodied in a sufficiently long 

 series of statistical records, establish that 



Prob. A=a, Prob. B=/5, 

 the very same experience may, by establishing also that 



Prob. AB=«ft 

 whence in conjunction with the former it follows that 



Prob. AB'=«/3', Prob, A'B = a'/3, Prob. A'B'=a'/3', 

 enable us to pronounce that A and B are in the long run, as to 

 happening or not happening, in the position of mutually inde- 

 pendent events. 



" 3rdly. I think it may be shown to demonstration, from the 

 nature of the result, that the solution you have obtained does 

 not apply simply and generally to the problem under the single 



