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LXXV. On Some Aspects of the Iheory of Probability. By 

 DuROTHY Wrinch, Lecturer at University College, London, 

 and Harold Jeffreys, M.A., D.Sc, Fellow of St. Johns 

 College, Cambridge *-. 



I. The Nature of Probability. 



1~^HE theory o£ probability suffers at the present time from 

 the existence of several different points of view, whose 

 relations to one another have apparently never been adequately 

 discussed. On the one hand some authorities follow de Morgan 

 and Jevons in regarding probability as a concept compre- 

 hensible without any definition, and perhaps indefinable, 

 satisfying certain definite laws the logical basis of which is 

 not yet clear. On the other hand, attempts have been made 

 to give definitions of probability in terms of frequency of 

 occurrence; of these one is due to Laplace, who was largely 

 followed by Boole, and another to Venn. Frequency of 

 occurrence being a well-understood mathematical concept, 

 such a definition would be important if it could be carried 

 out; for then the undefined notion of probability would be 

 expressed in terms of others that are better understood, and 

 its laws, if true, would become demonstrable theorems in 

 pure mathematics instead of postulates. Thus the subject 

 would acquire the certainty of any other portion of pure 

 mathematics and it would be unnecessary to investigate its 

 foundations independently. It appears, however, as we hope 

 to show in the first part of the present paper, that the 

 definitions offered either implicity involve the very notion 

 they are meant to avoid, or else make assumptions which are 

 actually erroneous. We therefore consider it best to regard 

 probability as a primitive notion not requiring definition. 



Laplace defines probability f as the ratio of the number of 

 favourable cases to that of all possible cases, and then goes 

 on to say " but that supposes the various cases equally 

 possible," so that to understand this definition it is necessary 

 to examine what Laplace meant by equally possible. The 

 expression is meaningless as it stands, for a proposition 

 relative to a set of data is always either possible or im- 

 possible ; there can be no degrees of possibility. He 

 indicates later that if a coin is unsymmetrical the probability 

 of throwing a head may be greater than that of throwing a 

 tail, though the difference may be small; yet both are 



* Communicated by the Authors. 



f Theovie analytique des probabilitis, troisieme Edition, p. 7 of iat in- 

 duction. 



