the Theory of Probabilities. 439 



2nd. B those combinations which cannot happen if w happen, 

 but may otherwise happen. 



3rd. C those combinations which may or may not happen if 

 w happen. 



4th. D those combinations which cannot happen at all. 



And the above representing all possible combinations, we have 



A + B + C + D = l (6) 



Now there are many problems in which the combination de- 

 noted by C does not present itself. Such is the one considered 

 by Mr. Cayley and myself, and commented on by Mr. Wil- 

 braham. As the principle of solution is the same in this as in 

 the more general class of problems in which C does appear, I 

 shall, for simplicity, confine myself to the simpler case. The 

 event w, then, consists solely of that combination of the simple 

 events s, t, &c. which is denoted by A, and the sole condition to 

 which those events are subject is 



D = 0, or A-fB = l; (7) 



these logical equations being, by virtue of the necessary equa- 

 tion (6), strictly equivalent when C does not make its appearance 

 in the development. 



The problem may now be briefly stated as follows. The events 

 s, t, &c. are subject to the condition (7), and at the same time 

 their respective probabilities are ^ 



Prob. s=Pj Prob. t=q, &c. 



Required the value of Prob. A. 



Now let us consider whether, upon the familiar notion of an 

 urn containing balls, we can construct a problem whose expressed 

 data shall be in all respects the same as the above, and which 

 shall at the same time admit of definite solution. 



And, in the first place, it is manifest that any event or combi- 

 nation of events may be represented by the issuing of a ball 

 possessing a particular quality, or combination of qualities from 

 an urn. Thus the event s may be represented by the issuing of 

 a ball possessing a particular quality which we will term the 

 s-quality, the event t by that of a ball possessing the Equality, 

 and so on. In like manner the event st, or the combination of 

 the events s and t, may be represented by the issuing of a ball 

 possessing at once the qualities s and t. And generally the 

 events A, B, D, whatever combinations of the symbols Sj t ,. 

 these letters may stand for, may be represented by the issuing 

 of balls possessing the corresponding qualities or combinations 

 of qualities. 



And as every species of events can thus be represented by the 

 issuing of a ball of a particular species from an urn, so every 



