MEASUREMENT OF VARIABLE PROPERTIES 119 



series is thus more satisfactory than between the series of 20 

 tosses. One series of 100 observations (tosses) with one coin 

 gives, therefore, more or less rehable information. 



In 6 series of 500 successive tosses the event a was observed 



235. 239, 248, 248, 262, 262 



times. The calculated frequency is ^%%. Here the concordance 

 is still more satisfactory than between the series of 100 tosses. 



The 3000 tosses being brought together into one series, the 

 observed frequencies were : heads 1494 times ; tails, 1506 times. 



The observed ratio is 0-996 : i. 



The calculated ratio is 1500 : 1500 = 1:1. 



The observed figures are as satisfactory as might be expected 

 from similar experiments. (Compare §§ 95 and 103.) 



§ 94.— THIRD EXAMPLE : THE POSITIONS OF EQUI- 

 LIBRIUM OF A SYSTEM OF TWO COINS. Let us toss 

 successively two coins 1. and II. Under the influence of chance 

 the first coin 1. may give the events a (head) or h (tail), which 

 we call from henceforth simple events. We ascribe to the letters 

 a and h, which represent the simple events, the numerical 

 value of their respective frequencies ; thus (according to § 92) 

 a = \ and& = | (or 0-50 and 0-50). 



In a similar way the second coin II. may give the simple events 

 a ox h (frequencies: a = 1=050 and & = 1 = 0-50). By the 

 combination of the simple events two by two (the number of 

 coins is two), four compound events may be brought about — viz. 



a (coin I.) followed by a (coin II.) 

 h „ „ „ h 



compound event aa 

 „ ah 



yy &« 



hh 



Since four possibilities (states of equilibrium, compound 

 events) exist, and since they are equally possible (stable), the 

 frequency of each compound event (calculated a priori) is i : 4. 

 (See § 92.) This is expressed by the numerical value of each 

 of the monomials which represent the compound events — viz. 

 aa-\\ ab = i, etc. 



In general, the frequency of any compound event is the product 

 of the frequencies of the simple events of which it consists. 



We have hitherto taken the order of succession of the simple 

 events into account. We may, however, neglect this order 

 and consider only the final result, as if both coins were tossed 

 at the same time. (By adopting this standpoint we do not 

 alter the facts.) We see then that there is no longer any dif- 

 ference between ah (head-tail) and ha (tail-head) ; both are 



