﻿Measurement of Chance. 73 



be compared by means of these definitions. Since we take 

 the meaning of chance to be that of the derived magnitude, 

 the definitions will be satisfactory if they are in accord- 

 ance with the derived process of measurement ; but we 

 shall not succeed in establishing an independent system of 

 fundamental measurement, unless the definitions are such 

 that they can be applied without resort to that process. 



The arbitrary assignment is usually made by attributing 

 the value 1 to the chance of an event which always happens 

 as the result of a trial. The only question that can arise 

 here, namely whether all other chances can be connected 

 with this chance by addition and equality, will be considered 

 presently. The definition of addition presents no difficult}'. 

 The chance of A happening is the sum of the chances of 

 .r, y, z, ... happening if, x, y, z, ... being mutually exclusive 

 alternatives, A is the event which consists in the happening 

 of either x or y, or z, .... This proposition is introduced in 

 all discussions of chance, but it is often introduced as a 

 deduction and not as a definition. The inconsistencies which 

 result from such a procedure are discussed in P. pp. 174, 184, 

 185. As a definition, it is satisfactory in our sense, for 

 measurements by the derived method would show that, in 

 such conditions, the chance of A is the sum of the chances of 

 x, y, z, ..., and yet it does not presuppose such derived 

 measurements. If the points in the derived measurement 

 lay on the straight line, this result would be a direct 

 consequence of the definition of the derived magnitude ; 

 but since they do not, it can be deduced from that 

 definition only if some assumption about the distribution 

 of the errors is made. The assumption that the errors are 

 random would probably suffice if randomness could be 

 strictly defined ; since it cannot, the agreement of the 

 proposed definition of addition with the results of the derived 

 process of measurement must be regarded as an experimental 

 fact. The definition is thus precisely analogous to that used 

 in the fundamental measurement of resistance, namely that 

 resistances are added when the bodies are placed in series. 



The definition of equality is much more difficult; in fact, 

 it is the stumbling block of many expositions of the measure- 

 ment of chance. For resistances we can say that bodies are 

 equal if, when one is substituted for another in any circuit, 

 the current and potentials in that circuit are unchanged ; 

 that definition does not involve a knowledge of Ohm's law 

 and of the derived measurement. The only attempt at an 

 analogous definition for chance, of which I am aware, is 

 that based on the principle of sufficient reason ; chances 



