Sec. 4.1 



THE BINOMIAL FREQUENCY DISTRIBUTION 



79 



If 10 persons are insured under conditions to which the American 

 Experience Mortality Table applies, it follows from the discussion 

 of section 3.3 that the 11 corresponding probabilities of occurrence 

 of these numbers of deaths are given by the successive terms of the 

 following binomial series: 



(.9 + .l)io = (.9)10 + 10 (.9)9(.i)i + 4 5(.9)8(.i)2 + ... +(-1) io 



It is not necessary to devise some game with p = .1 and discover from 

 experience that a fraction (.9) 10 of the trials will show no occurrences 

 of the event E because the only interest is in the relative number of 



Figure 4.11. Graph of the binomial frequency distribution for p = 1/2 and 



n = 10. 



occurrences, and that is what the probability gives. Hence, the above 

 series gives the frequency distribution of the eleven possible classes 

 of events. 



To re-illustrate the discussion of the preceding paragraphs with an 

 example which the reader can reproduce easily and, in addition, to 

 show how to graph a binomial frequency distribution, attention again 

 is called to a mathematical model. Suppose that an unbiased coin 

 is to be flipped 10 times and the number of heads is to be recorded 

 after each set of 10 throws. In these circumstances, n — 10, p = 1/2, 

 and q = 1 — p = 1/2; hence the successive terms of the following 

 binomial series give the probabilities for 0, 1, 2, 3, . . . , or 10 heads 

 on any future set of 10 throws: (1/2) 10 + 10 (1/2) 9 (1/2) * + 45(l/2) 8 

 (1/2)2+-.-+ (1/2) 10 ; or 1/1024 + 10/1024 + 45/1024 +••• + 

 1/1024. In view of the fact that each of the denominators is 1024, 

 we obtain a useful and simpler expression for the relative frequency 

 of occurrence of 0, 1, 2, ... , or 10 heads on 10 throws, by using only 

 the numerators. From them a graph can be constructed to depict 

 the relative frequency for each possibility, as is done in Figure 4.11. 



