124 THE QUANTITATIVE METHOD IN BIOLOGY 



cause which we call chance is indeed a combined cause which 

 consists of an unlimited (unknown) number of simple causes. 



§ 98.— SIXTH EXAMPLE : AN URN CONTAINING loo 

 WHITE AND 100 BLACK BALLS.— When one baU is ex- 

 tracted at random from the urn, two simple events are possible : 

 white = a OT black = b. Since the number of cases (possibilities) 

 favourable for a is loo and the total number of possible cases 

 200, the frequency of a is loo : 200 = J. The frequency of b 

 has, of course, the same value. The conditions are thus 

 exactly the same as in the example of one coin (§ 92) . If the 

 balls are extracted by series of 2, 3 ... w, taking them one by 

 one and putting each ball back into the urn after the colour has 

 been ascertained,^ the same compound events (combinations) 

 will occur as when 2, 3 ... w coins are tossed. 



If the balls are extracted (one by one), for instance, in series 

 of eight, all the possible combinations and the frequency of 

 each of them may be calculated a priori by expanding (a + by, 

 in which a = b = l — viz. 



(a-hb)^ = a^ + 8a'b + 2Sa^b^ -f S^a^b^ + yoa^b^ + 56^3^5 ^ gSa^i^ + 

 8ab' + b^ = i (certitude). 2 



QUESTION I. : What is the frequency of the compound event 6 white 

 (a) and 2 black (6) balls ? 

 Answer : Approximately 28 : 256 ( = o'io9). 

 QUESTION II. : Which is the frequency of aababbab ? 

 Answer : Approximately i : 256 = 0004. 



§ 99.— SEVENTH EXAMPLE : AN URN CONTAINING 

 200 WHITE AND 100 BLACK BALLS.— When one baU is 

 extracted the frequency of a (white) is %%% (number of favour- 

 able cases divided by number of possible cases) or |, and the 

 frequency of b (black) is |. All possible information is obtained 

 by expanding {a + b^, in which n is the number of extracted 

 balls (taken one by one ; see the note, p. 124), a=% and 6 = J. 

 (See (i)-(7), p. 120.) 



QUESTION I. : Six balls being extracted, which is the frequency of the 

 compound event two white and four black balls ? (See (a + 6)^ p. 123.) 



Answer : Approximately 60 : 729 (= 0*082 ). 



QUESTION II. : 15,000 series of eight balls being extracted, how many 

 times has the combination seven white and one black ball been obtained ? (See 

 (a + bf, p. 124.) 



A nswer : 15,000 x 8a'''6 =15,000 x ^^f = approximately 2341 times. 



1 If the ball were not put back into the urn after each extraction, the fre- 

 quency of the simple events would be continually modified and the calculations 

 would become more complicated. 



Let us suppose that the first extracted ball is a. When a second ball is 

 taken, the frequency of the event a is no longer ig§, but ^■^, and the fre- 

 quency of the event b is |§§, etc. 



2 Sum of the coefificients = 2^=256. 



