636 PROFESSOR BOOLE ON THE COMBINATION 
as the only values which satisfy (12). Hence 
ce” 
Probability sought = gs ae 
Sg ee ae ee? NR 
This result is evidently correct. The probability that such an event will take 
place when two other events, # 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 a is never absent. 
5thly, If e=e' and g=1—p, we find in like manner w=t, whence 
Probability sought = 2 ight) | | ea ori) 
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 2 
6thly, But if g=1—p, while cand ¢’ are not equal, then the value of the probability 
sought is no longer 5 It may be shown, by a proper discussion of the formule, 
that the presumption afforded by the event 2, 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 7 in conjunction 
with the event « is due to the efficient action of the event «, or whether it is a 
product of some other cause or causes. The more frequent the occurrence of 2, 
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, w were 
a standing event, or a state of things always present, the probability that any 
event z would occur when w 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. 
7thly, The case in which e=¢ and p=g, is a very interesting one. A careful 
analysis leads to the following results. 
If there be two events ¢ and y, which are in themselves equally probable, the 
probability of each being c, and if when the event w is known to be present, while 
