﻿Measurement of Chance. 71 



by chance, then it is true that we need only plot one point 

 on the line ; and the distribution of the "errors is such that 

 the relative error of a determination from a single point is 

 less the greater the number of trials involved. We shall 

 group all our observations together, so as to make their 

 total number as great as possible. But similarly, if we know 

 that the material of the die obeys Ohm's law, one observa- 

 tion is sufficient to determine its resistance ; and the accuracy 

 of the determination will be greatest if we choose the 

 measuring current within a certain range. An even closer 

 parallel would be obtained if we took in place of resistance 

 the derived magnitude, uniform velocity. If we knew that 

 the velocity was uniform, we should choose our time and 

 distance as great as possible, and determine the velocity from 

 this single pair of values without troubling to plot smaller 

 values. 



But in order that determination by a single point should 

 be legitimate, we must know that the events really are 

 determined by chance, and the only test of chance is that, 

 when a series of points are plotted in the manner described, 

 the only regularity discoverable in them is that they lie 

 about the straight line. Their distribution about that line 

 must be random. Thus, to take Poincare's excellent example, 

 if the trials were made by selecting the first figure of the 

 numerals in a table of logarithms in the conventional order, 

 and the events were the occurrence of the figure 1, the 

 plotted points would lie on the whole about a line with 

 a slope of 1/10. But a regularity of distribution about 

 that line would be apparent ; we should have a series of 

 points all lying above the line followed by a set all lying 

 below. If, on the other hand, we took the last figure of 

 the numerals, no such regularity would be apparent ; the 

 distribution of the points about the line would be random ; 

 the events would be dictated by chance. 



It is of the first importance to insist that in measuring a 

 chance we are picking out the only regularity that we can 

 find in some sequence of phenomena, leaving a residuum 

 which is purely random. Randomness is a primary con- 

 ception, incapable of further definition ; it cannot be explained 

 to anyone who does not possess it. It is based, I believe, on 

 observation of the actions of beings acting consciously under 

 free volition ; and it is subjective in the sense that what is 

 random to one person may not be random to another with 

 fuller knowledge (P. p. 203") . There are certain forms 

 of distributions that are random to everybody ; it is this 

 common randomness, objective in the sense in which all the 



