Oh the Foundations of the Theory of ProhahUities. By R. L. Ellis, Esq. 

 M.A., Fellow of Trinity College. 



[Read Feb. 14, 1842.] 



The Theory of Probabilities is at once a metaphysical and a mathematical science. The 

 mathematical part of it has been fully developed, while, generally speaking, its metaphysical 

 tendencies have not received much attention. 



This is the more remarkable, as they are in direct opposition to the views of the nature 

 of knowledge, generally adopted at present. 



(2.) The theory received its present form during the ascendancy of the school of Con- 

 dillac. It rejects all reference to a priori truths as such, and attempts to establish them as 

 mathematical deductions from the simple notion of probability. Are we prepared to admit, 

 that our confidence in the regularity of nature is merely a corollary from Bernouilli's theorem ? 

 That until this theorem was published, mankind could give no account of convictions they had 

 always held, and on which they had always acted .'' If we are not, what refutation have we to 

 give .'' For these views are entitled to refutation, from the general reception they have met 

 with, from the authority of the great writers by whom they were propounded, and even from 

 the imposing form of the mathematical demonstration in which they are invested. 



I shall be satisfied if the present essay does no more than call attention to the inconsist- 

 ency of the theory of probabilities with any otlier than a sensatio7ial 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 possible ways in which it may take 

 place, divided by the total number of such ways which may occur on the given trial. 



If a,, 6,, TWi, denote equally possible cases which may occur on one trial, a.;,b2,...k2 those 



which may occur on a second trial, 0363. ...pa those belonging to a third. Sec: then a,, 

 62O3.... aia^bfi...hc. &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 



'-^^^^, rm' (1 - w)*-^ : 



l.Z...pl.Z...{k-p) 



this follows merely by the doctrine of combinations. These are all the propositions to which 

 I shall have occasion to refer. 



(■1.) If the probability of a given event be correctly determined, 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 

 Vol. VIII. Paet I. A 



