362 Mr. A. Cayley on a Question in the 



modification of the assumption that A and B are independent. 

 The completed data under this assumption are 



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



Prob. AE = «p, Prob. BE=/%. 



You may deduce all these from your Table of probabilities of 

 ' compound events ' given in your paper. Now you may easily 

 satisfy yourself that the sole necessary and sufficient conditions 

 for the consistency of these data are the following : — 



(1) «p> + /3q = a /3. 



(2) «?+/%'= a/3. 



(3)1 



p 



I. 



0. 



(M) 



But your solution requires the following conditions to be satisfied, 

 viz. 



q-ap = 0, p-fiq = 0, 



together with the system (3). Now (1) and (2) are expressible 

 id the form 



From which you will see that your conditions are narrower than 

 those which the data are really subject to. If your conditions 

 are satisfied, the data will be consistent ; but the converse of this 

 proposition does not hold. 



" 4thly. You remark that my solution of the problem, in which 

 the independence of A and B is not assumed, but in which the 

 probabilities are otherwise the same as in yours, is only appli- 

 cable when 



a' + ctp=l3q, /3' + /3q = ap; 



but you do not appear to have noticed that these are actually 

 the conditions of consistency in the data. Unless these are satis- 

 fied, the data cannot possibly be furnished by experience. 



" 5thly. You remark that I have solved the problem under what 

 you call the ' concomitance ' statement, and not the c causation 3 

 statement. I think that every problem stated in the ( causation' 

 form admits, if capable of scientific treatment, of reduction to the 

 1 concomitance ' form. I admit it would have been better, in 

 stating my problem, not to have employed the word ' cause 3 at 

 all. But the introduction of the hypothesis of the independence 

 of A and B does not affect the nature of the problem. 



