Theory of Probabilities. 353 



the question : viz., if for shortness we use AE to denote the com- 

 pound event A and E, and so in other cases ; and if we use also 

 A' to denote the non-occurrence of the event A, and so in other 

 cases (of course (AE)', which denotes the non-occurrence of the 

 event AE, must not be confounded with A'E', which would de- 

 note the non-occurrence of each of the events A, E), then the 

 question is, " Given 



Prob. A'B'E, = 0, 

 „ A , = a, 

 „ AE, = up, 



,i B , = ft 



„ BE, = fa) 

 required the probability of E." To show that the two state- 

 ments are really distinct questions, it may be observed that when 

 A and B both exist, then, according to the causation statement, 

 they may one or each of them act efficiently, and E may thus 

 happen as an effect of one of them only, or as an effect of each 

 of them ; but, according to the concomitance statement, E can- 

 not be attributed rather to one of the events A, B, than to the 

 other of them, or to both of them. 



The solution which I gave in the year 1854 (Phil. Mag. 

 vol. vii. p. 259) refers to the causation statement of the question, 

 and assumes the independence of the two causes*; and on this 

 assumption I believe it to be correct. And I remark, in passing, 

 that in the strictest sense of the word cause, all causes are ex vi 

 termini independent. The solution was as follows: — Let u be 

 the required probability; Xthe probability that A acting, it will 

 act efficiently j //, the probability that B acting, it will act effici- 

 ently ; then we have 



u — Xa + fifS — XfjLct/3, 



p=X +(1-\)m/5, 



q= fi +(l—fi)\ct; 



and eliminating \, \x from these equations, we have the required 

 probability u. 



As I did not further work out the solution, I omitted to state 

 the relations of inequality presupposed among the data a, j3, p, q, 

 or to show how the sign of the quadratic radical in the resulting 

 expression for u was to be fixed. The omissions in question 

 were supplied by Dr. Dedckind, in his paper " Bemerkungen zu 

 einer Aufgabe dec Wahrscheinlichkeitsrechnuug," Crelle, vol. 1. 



* It is part of the assumption, that the causes do not combine to produce 

 the effect : viz. if they both act, the effect is not produced unless one of 

 them acts efficiently ; thev may or may not each of them act efficiently. 



Phil. Mag. S. 4. Vol. 23. No. 155. May 1862. 2 B 



