78 BINOMIAL AND NORMAL DISTRIBUTIONS Ch. 4 



4.1 THE BINOMIAL FREQUENCY DISTRIBUTION 



As was stated above, a binomial population and the corresponding 

 binomial frequency distribution are involved when every single event 

 which can occur under prescribed conditions must belong to one of 

 two classifications. This fact corresponds to the meaning of the 

 prefix bi- in the word binomial. For example, if you take out a term 

 insurance policy for a period of 10 years you and the company are 

 interested in your subsequent classification as "dead" or "alive" 

 before, or at, the end of the 10 years. Of course, the company in- 

 sures many persons and regards them as a group, some of whom 

 will be classifiable as "dead" and the remainder as "alive" at the 

 expiration of the 10-year term. What the insurance company and 

 its clients need to know, then, is this: Given a group of n persons 

 insured for a 10-year term, what are the probabilities associated with 

 each of the possible numbers of "dead" and "alive" insured persons 

 during the 10-year period of the insurance contract? For any spe- 

 cific n the relative frequency — over a great deal of experience — of the 

 occurrence of 0, 1, 2, 3, ... , (n — 1), or all n "dead" after 10 years 

 will be the binomial frequency distribution mentioned above. It is 

 upon the basis of this sort of information that insurance premiums 

 are calculated. 



Suppose, for simplicity, that a company has insured 10 persons 

 who are 30 years of age for a 10-year term. What can the company 

 expect to pay out in death benefits? It is obvious that at the end 

 of the 10-year period any one of 11 events may have occurred. There 

 can be 0, 1, 2, ... , or all 10 classified as "dead." Also, over the 

 experience of many such groups of contracts for 10-year periods those 

 11 possible outcomes will occur with unequal relative frequencies 

 which depend both on the number, such as n = 10, and on the prob- 

 ability of death for persons in this age interval. Clearly, this bi- 

 nomial frequency distribution depends on n and on p = probability 

 of death between the ages of 30 and 40 years. 



No one can state theoretically what the probability of death is for 

 any particular person during the age period of 30 to 40 years; but 

 tables have been compiled from experience which give the best avail- 

 able estimate of the desired probability. For example, the American 

 Experience Table of Mortality indicates that the average probability 

 is approximately one-tenth that an insurable person (determined by 

 examination before the company will insure) now 30 years of age 

 will die before he is 40 years of age. 



