XXVII. Remarks on the Fundamental Principle of the Theory of Probabilities. 

 By R. L. Ellis, M.A. late Fellow of Trinity College. 



[Read Nov. 13, 1854.] 



I wish to make an addition to the remarks on the foundation of the theory of probabilities 

 which were offered some years since to the notice of the Society*. My intention in doing 

 so is to consider, in what way the proposition, which I conceive to be the fundamental 

 principle of the theory, may be the most clearly and conveniently expressed. This principle 

 may for the moment be thus stated : " On a long run of similar trials, every possible event 

 tends ultimately to recur in a definite ratio of frequency." Our conviction of the truth of this 

 proposition is, I think, intuitive, — the word being used, as in all similar cases, with reference to 

 the intuitions of a mind, which has fully and clearly apprehended the subject before it, and 

 to which therefore to have arrived at the truth and to perceive that it has done so are 

 inseparable elements of the same act of thought. If we endeavour to translate the propo- 

 sition just stated into ordinary philosophical language, we may in the first place remark that the 

 phrase " similar trials," expresses the notion of a group or genus of phenomena to which 

 the different results are subordinated as distinct species. If the trial is the throwing of a die, 

 this may be regarded as the generic character; the occurrence of ace, deuce, &c. consti- 

 tuting different species. Thus much is clear ; but it is less obvious how the idea expressed 

 by a " long run of trials" in " definite series of experiments, 1 ' and the like, is to be expressed, 

 so as to make the analogy between the fundamental principle of the theory of probabilities 

 and those of other sciences more obvious than it has hitherto been. The idea in question 

 is not readily expressed in any way, because in its own nature it is negative and indefinite. 

 The phrases I have just quoted imply merely the absence of the limitations inseparable 

 from individual cases, or from any finite number of such cases, whether contemplated as 

 actually existent or as about to be developed within definite limits of space and time. 



When individual cases are considered, we have no conviction that the ratios of frequency of 

 occurrence depend on the circumstances common to all the trials. On the contrary, we recog- 

 nise in the determining circumstances of their occurrence an extraneous element, an element, 

 that is, extraneous to the idea of the genus and its species. Contingency and limitation come 

 in (so to speak) together ; and both alike disappear when we consider the genus in its entirety, 



• Transactions of the Cambridge Philosophical Society, Vol. VIII. page 1. 



