﻿Measurement of Chance. 75 



is not true, if the chances are always experimentally deter- 

 mined. Consider, for example, the disintegration of a 

 radioactive atom within a stated period. There are only 

 two alternatives : the atom does disintegrate, or it does not. 

 The sum of the chances of these two events is 1, but the 

 chances of the two are not in general equal. And neither 

 of them can be shown experimentally to be the sum of the 

 equal chances of other events such that the sum of some 

 other set of those chances is equal to the unit. The definition 

 of unit chance together with definitions of equality and 

 addition would never permit us to determine such chances ; 

 they can only be determined by derived measurement. 



9. Chance is therefore not capable of fundamental 

 measurement. Nevertheless the principles of fundamental 

 measurement are important in connexion with chance, 

 because they are involved in the calculation of chances. 

 When we calculate a chance we always assume that it is 

 measurable by the fundamental process. Thus if we calculate 

 the chance of drawing a heart from a pack, we argue thus : — 

 The chance of drawing any one of the 52 cards is equal to 

 that of drawing any other. The chance of drawing one of 

 the 52 cards is, by the definition of addition, the sum of the 

 chances of drawing the individual cards, and, by the 

 definition of unit chance, it is 1. Consequently the chance 

 of drawing any one card is 1/52. But the chance of drawing 

 a heart is the sum of the chances of drawing 13 individual 

 cards ; it is therefore the sum of 13 chances each equal to 

 13/52, i. e. 1/4. The calculation is perfectly legitimate, so 

 long as we know (1) of how many individual events the 

 event under consideration (and any other event introduced into 

 the argument) is the sum, and (2) that the chances of these 

 individual events are equal. (1) does not depend on the 

 derived system of measurement, but it does involve a very 

 complete knowledge of the event under consideration; (2), if 

 it is an experimental proposition at all, must depend upon 

 derived measurement. The calculation is often made when 

 (2) is not experimental, and when there is no direct know- 

 ledge of (1) ; it is then purely theoretical, and the only 

 legitimate use that can be made of it is to confirm or reject 

 the theory by means of a comparison of the calculated chance 

 with that determined experimentally by the derived measure- 

 ment. The fact remains that true chance, the property of 

 the system, is always and inevitably measured by the derived 

 process and not by the fundamental. 



