632 PROFESSOR BOOLE ON THE COMBINATION 



well as of z. A circumstance is an event — a state of things which comes to pass, 



or has come forth — evenit. 



The data leave wholly arbitrary the probabilities of the event x and y. Thus 



p and q are conditional probabilities ; p is the probability that if the event * 



occur, the event z will occur ; q is the probability that if the event y occur, the 



event z will occur. Hence 



_Prob. ocy _Prob. yz 



P ~ Prob. aT' q ~ VvobTy ' • • • I 1 ) 



Our object is to determine the probability that if the events * and y both occur, 

 the event z will occur. We have therefore to seek the value of the fraction 



Prob. ccyz 

 Prob. ocy 



or, as for our present purpose it is more convenient to say, of 



Prob. xyz ^ ^ 



Prob. xyz + Prob xyz 



In seeking the value of Prob. xyz, which we shall represent by u, the formal 

 statement of our data and qucesitum will therefore be 



_. f Prob. x = c, Prob. ?/ = c' ) 



Given I J , . . . . (3) 



( Prob. xz = cp, Prob. yz = cq J 



Required Prob. xyz. 



c and c' being arbitrary constants expressing the unknown probabilities of the 

 events x and y. 



A misconception may here arise respecting the meaning of Prob. x, Prob. y, 

 which it is worth while to anticipate. In the case of testimony, Prob. x would 

 not mean the probability that a testimony would be borne, but the probability 

 that the particular kind of testimony actually recorded considered with reference 

 to its object, credibility, &c, would be borne. Testimonies differ, not merely as 

 to their degree of credibility, but as to their unexpectedness — as to the surprise 

 which they occasion. And it is, I think, matter of personal experience that this 

 unexpectedness is in itself an element affecting the strength of that expectation 

 which combined testimonies produce. So, too, if x and y are facts of observation, 

 e.g., observed symptoms of a disease z, the probability of that disease, when both 

 symptoms present themselves, is not determined by the strength of the separate 

 presumptions merely, but is consciously increased by our knowledge of the rarity 

 of the symptoms themselves. And thus the elements Prob. * Prob. y, which 

 have been introduced by a formal necessity of the statement of the problem are 

 seen to belong to the very matter of its solution. 



Making 



xz—s, yz=t, xyz — (f>, 



