The binomial distribution

The binomial distribution is the probability distribution of compound events (section 3.6) consisting of the joint occurrence of independent simple events (sections 3.9 and 15.3). Each simple event may take two complementary forms (section 3.7) of which the probabilities are p and q respectively. Because the two forms which each simple event may take are complementary,(p + q) =1. Moreover, since the simple events are independent from each other, the probabilities of the various kinds of compound events may be obtained by adding up the probabilities of the two forms and by multiplying the probabilities of the simple events, which yields $$ \begin{gathered} \left( {p + q} \right)\left( {p + q} \right) .... \left( {p + q} \right) = \left( {p + q} \right)^k = 1. \hfill \\ \underbrace {1^{st} 2^{nd} .... k^{th} }_{simple events} \hfill \\ \end{gathered} $$ In order to make the discussion of the binomial distribution concrete and easy to follow, the example of the frequency of male and female children in human families is intro-duced without delay. Each simple event here is the birth of a child and, since children of ill-determined sex are very rare, it may be considered in practice that the birth of a boy, of which the probability is p, and the birth of a girl, of which the probability is q, are complementary events, whence (p + q)= 1. The compound event is the occurrence of a family of kchildren of whom Xare male and (k-X) are female. Except in the case of true (monozygotic) twins, who always have the same sex, the sex of simultaneous or successive children is thought to be independent and the probability of each type of family may be determined by multiplying each term of the binomial corresponding to the first birth by each term of the binomial corresponding to the second birth and so on. Because of the current knowledge about the mechanism of sex determination, and because the relative frequencies of male and female newborns are usually very close, it is generally considered that p=q=0.5, but the analysis of large samples may give indications that the probability of the birth of a boy differs slightly from that of a girl. The relative frequency of male and female children in human families is an excellent example of the binomial distribution and will be used through this chapter.