ON THE FOUNDATIONS OF 



I shall be satisfied if the present essay does no more than 

 call attention to the inconsistency of the theory of probabilities 

 with any other than a sensational philosophy. 



3. As the first principles of the mathematical theory are 

 familiar to every one, I shall merely recapitulate them. 



If on a given trial, there is no reason to expect one event 

 rather than another, they are said to be equally possible. 



The probability of an event is the number of equally pos- 

 sible ways in which it may take place, divided by the total num- 

 ber of such ways which may occur on the given trial. 



If tfj, 5j, m 1? denote equally possible cases which may 



occur on one trial, a 2 b 2 \ those which may occur on a 



second trial, 3 5 3 .p 3 those belonging to a third, &c. : then 



afaa^ ! 2 ^ 3 &c. &c. are all equally possible complex 



results. 



Hence it follows that on the repetition of the same trial 

 k times, the probability that an event whose simple probability 

 is m will occur p times is 



1 .2. ..pi .2...(k-p) 



this follows merely by the doctrine of combinations. These are 

 all the propositions to which I shall have occasion to refer. 



4. If the probability of a given event be correctly deter- 

 mined, the event will, on a long run of trials, tend to recur with 

 frequency proportional to this probability. 



This is generally proved mathematically. It seems to me 

 to be true a priori. 



When on a single trial we expect one event rather than 

 another, we necessarily believe that on a series of similar trials 

 the former event will occur more frequently than the latter. 

 The connection between these two things seems to me to be an 

 ultimate fact, or rather, for I would not be understood to deny 

 the possibility of farther analysis to be a fact, the evidence of 

 which must rest upon an appeal to consciousness. Let any one 

 endeavour to frame a case in which he may expect one event on 

 a single trial, and yet believe that on a series of trials another 

 will occur more frequently ; or a case in which he may be able 



