120 THE QUANTITATIVE METHOD IN BIOLOGY 



confounded, and only three different compound events ought to 

 be distinguished — viz. 



aa ah + ha bb 



or 



a^ + zab + &2 



These three compound events have not the same frequency. 

 Each of the events a^ and ¥ is produced in one way ; its fre- 

 quency is \. The event ab may be produced in two 'ways [ab 

 or ba), each of which has the frequency \. The frequency of 

 ah is the sum of the frequencies of these two ways — ^viz. \ + \ = \. 



The trinomial a^ + 2ab + b^ affords us complete information 

 about the events which may occur when two coins are tossed 

 once — viz. 



(i) Each letter represents a simple event (two simple events 

 are in play). 



(2) The numerical value of each letter represents the fre- 

 quency of the corresponding simple event (a = | and 6 = J). 



(3) From the number of terms we know how many sorts of 

 compound events are possible (three). 



(4) The nmnerical value of each term represents the fre- 

 quency of the corresponding compoimd event (for instance, the 

 frequency of ah is zab = f = J) . 



(5) The sum of aU the coefficients ^ is the number of possible 

 compound events, the order of succession being taken into 

 account (1 + 2 + 1 = 4). 



(6) The letters of each term represent the constitution 

 (simple events) of the corresponding compound event. 



(7) The coefficient of each term indicates in how many ways 

 the corresponding compound event may be brought about (for 

 instance, ab in two ways : ab and ba). 



All the above information may be obtained in the following 

 way: — 



The binomial a + b or {a + by (in which a = b = l) represents 

 the events which may be observed when one coin is tossed. 

 When two coins are tossed we obtain a complete representation 

 of all the events by multiplying the binomials of both coins into 

 one another — ^viz. 



(a + b) x{a + b) = {a + b)^ = a^ + 2ah + b" 



All the above may be verified approximately by observation, 

 under the condition that the number of experiments (tosses) is 

 large enough. Example : If two coins are tossed 3037 times, 

 the event ab may be expected approximately 1518-5 times 

 (1518 or 1519 times), and each of the events a^ or b^ approxim- 

 ately 759 '25 (759) times. 



1 Or the denominator of the numerical value of any term. 



