TtlEASUREMENT OF VARIABLE PROPERTIES 127 



Table p 



B 



I.I 1.2 1.3 1.4 1.5 1.6 



2.1 2.2 2.3 2.4 2.5 2.6 



3.1 3.2 3.3 3.4 3.5 3.6 



4.1 4.2 4.3 4.4 4.5 4.6 



5.1 5.2 5.3 5.4 5-5 5.6 



6.1 6.2 6.3 6.4 6.5 6.6 



D 



In Table /3 the thirty-six compound events are represented 

 by their facial values ; in each pair of figures the first figure 

 represents the first die. In each of the eleven obUque rows 

 parallel to the diagonal BC the compound events have the same 

 facial value. 



The frequency of the eleven facial values is expressed by the 

 sum of the arithmetical values of the monomials which belong 

 to the same vertical column in Table a. For instance, the fre- 

 quency of 4 (4 dots with 2 dice) is 2ac -\-h'^ = -^ + -^ = ^.1 



In Table 7 the eleven facial values are given with their 

 respective frequencies : 



Fac. value 2 



Table y 

 '=; 6 



8 9 10 II 12 

 rrequency -sir ^^ 3F ^^ ^^ se- tts" -se" "so- st -sir 



The frequency increases regularly from 2 to 7 and decreases 

 further till 12. (Compare Table p.) This regular result 

 (which may be verified by experiment ; see § 103) is brought 

 about by chance ! The variation of the facial values is in- 

 cluded between two limits : 2 and 12. 



If n dice are cast the frequency and the facial value of all the 

 possible events may be obtained by expanding {a+h + c^-d^ 

 e + 1Y. 



The same method is applicable, of course, whatever may be 

 the number of faces of the dice (tetrahedral, octahedral, 

 dodecahedral, icosahedral dice, etc.) and the number of dice in 

 each cast. A coin may be looked upon as being a die with two 

 faces. In the experiments with coins (§ 94 ; see also § 98) all 

 the possibilities are latent in the expression {a-vb)^ '. this is a 

 pecuhar case of the general method expounded in the present 

 paragraph. 



§ 102.— TENTH EXAMPLE : ONE CAST WITH TWO 

 DICE : DIVERSITY OF THE EFFECTS.— In the preceding 

 example we consider a first series of six events (first die) which 

 coincide with the facial values i to 6, and a second similar series 



1 In other words, the facial value 4 is obtained in three different ways, each 

 of which has a frequency ^. (See Table /8.) 



