636 PROFESSOR BOOLE ON THE COMBINATION 



as the only values which satisfy (12). Hence 



Probability sought = - — - — = g (6) 



This result is evidently correct. The probability that such an event will take 

 place when two other events, x and y, are present, is the same as the probability 

 that it will take place when the event y is present, if it is known that the other 

 event x is never absent. 



otkly, If c=c' and q=l—p, we find in like manner u = t, whence 



Probability sought = h (7) 



This result is evidently correct. If the events or testimonies x and y are 

 equally likely to happen, and if the first yields the same presumption in favour 

 of that event whose probability is sought as the other yields against it, the 



chances are equally balanced, and the probability required is ^. 



6thly, But if q = 1 — p, while c and c' are not equal, then the value of the probability 



sought is no longer „• It may be shown, by a proper discussion of the formulae, 



that the presumption afforded by the event x, whether favourable or unfavour- 

 able, is stronger than the opposite presumption afforded by the event y, when- 

 ever c is less than c', and vice versa. And hence it follows, that if there be two 

 events which, by themselves, afford equal presumptions, the one for and the other 

 against some third event, of whose probability nothing more is known, then, if 

 the said two events present themselves in combination, that one will yield the 

 stronger presumption, which is itself, of the more rare occurrence. This, too, is 

 agreeable to reason. For in those statistical observations by which probability is 

 determined, we can only take account of co-existences and successions. We do 

 not attempt to pronounce whether the presence of the event z in conjunction 

 with the event x is due to the efficient action of the event x, or whether it is a 

 product of some other cause or causes. The more frequent the occurrence of x, 

 the less entitled are we to assert that those things which accompany or follow 

 it derive their being from it, or are dependent upon it. If, for instance, x were 

 a standing event, or a state of things always present, the probability that any 

 event z would occur when x and y were jointly present, would be the same 

 as the simple probability of that event z when y was present, and it would be 

 wholly uninfluenced by the presence of x. This is the limiting case of the gene- 

 ral principle. 



7tklt/, The case in which c=c' and p = q, is a very interesting one. A careful 

 analysis leads to the following results. 



If there be two events z and y, which are in themselves equally probable, the 

 probability of each being c, and if when the event x is known to be present, while 



