MEASUREMENT OF VARIABLE PROPERTIES 121 



REMARK : In the examples hitherto mentioned we have 

 at our disposal two methods for discovering the effects of the 

 variable combined cause which is called chance : (i) calculation 

 a priori, based upon the properties (specific energy) of the 

 objects under consideration (coins) ; (2) observation of facts 

 in sufficient number. 



§ 95.— FOURTH EXAMPLE : THE POSITIONS OF 

 EQUILIBRIUM OF THREE COINS.— Three coins being 

 tossed successively, the possible compound events are eight in 

 number — viz. : 



aaa aab abb bbb 

 aba bab 

 baa bba 



The order of succession being neglected (the coins being 

 tossed simultaneously), the eight compound events are reduced 

 to four. Taking the binomial of each coin (in which a = b = i), 

 complete information is obtained in the following way : — 



(a + b) x(a + b) x (a + b) = {a+b)^ = 

 = a^ + 3a^b + 3ab^ + b^ 



From this polynomial we obtain similar information to that 

 found in the trinomial in the third example (p. 120, (i)-(7)). 



Experimental verification: I have tossed three coins (British 

 pennies) simultaneously 1000 times. Result : 



First Series : 500 Successive Tosses 



Second Series : 500 Successive Tosses 



Compound 

 Events 



68 



Observed 

 Frequency 



47 



216 

 62 



Calculated 

 Frequency 



500 X 

 500 X 

 500 X 



= 62 5 



=187-5 



=187-5 



500 xi= 62-5 



