MEASUREMENT OF VARIABLE PROPERTIES 125 



§ 100.— EIGHTH EXAMPLE : AN URN CONTAINING 

 100 WHITE, 100 BLACK AND lOO RED BALLS.— When one 

 ball is extracted three simple events are equally possible : 

 a (white), h (black) and c (red). The frequency of each of them 

 is M§ = ^- This is represented by the trinomial a-\-b + c, or 

 1 + ^ + ^ = 1 (certitude). 



When two balls are extracted successively the possible events 

 with regard to the second ball are similarly represented by 

 a + b + c. Since the simple events are combined two by two, 

 nine compound events are (equally) possible — viz. 



aa bb cc ab ac be 

 ba ca cb 

 The frequency of each compound event is ^x J=^. 

 The order of succession being neglected, the nine events are 

 reduced to six. The expression of these six compound events 

 is obtained by multiplying into one another the trinomials 

 representing the simple events which are combined (see § 94, 

 p. 120) — viz. 



(a + & + c) X (a + 6 + c) = (a + 6 + c)^ = 

 a'^ + b'^ + c^ + 2ab + 2ac + 2bc = f = i 



The frequency of each compound event is expressed by the 

 nimierical value of its term. 



Complete information about aH the possible compound events 

 when series of 3, 4 ... « balls are extracted is obtained by 

 expanding (a + b + c)", in which n is the number of simple events 

 (extracted balls of one series) and a = b = c=\. (Continued in 

 § II5-) 



§ 101.— NINTH EXAMPLE : ONE DIE, TWO DICE.— In 

 the preceding examples the effects of chance are Umited to a 

 definite number of simple events. In each event a certain 

 simple property (head, tail ; colour) of the objects is rendered 

 visible. The simple events and properties are combined (co- 

 exist) in various ways, according to the rules of chance. 



Between the above-mentioned simple properties (primordia, 

 see § 92, p. 117, note 2) no quantitative relations exist. 

 Each primordium is distinctly different from the others ; it 

 is present or absent, visible or latent (concealed), without 

 more. 



It may happen, however, that the properties under con- 

 sideration are measurable, each simple event coinciding with 

 a certain value. In this case the rules of chance are un- 

 altered, but chance has certain effects which we have not yet 

 alluded to. 



One ordinary die being cast, six states of equilibrium or 

 simple events are equally possible : in each event one of the 



