Chapter 3, Problem 3.3TE

Consider a school community of m families, with $n_i$ of them having i children, $i=1,\mathrm{...},k,{\displaystyle \underset{i=1}{\overset{k}{\sum }}{n}_i}=m$. Consider the following two methods for choosing a child:

A. Choose one of the m families at random and then randomly choose a child from that family.

B. Choose one of the ${\displaystyle {\sum }_{i=1}^{k}i{n}_i}$ children at random.

Show that method 1 is more likely than method 2 to result in the choice of a firstborn child. Hint: In solving this problem, you will need to show that ${\displaystyle \underset{i=1}{\overset{k}{\sum }}i{n}_i}{\displaystyle \underset{j=1}{\overset{k}{\sum }}\frac{n_j}{j}}\ge {\displaystyle \underset{i=1}{\overset{k}{\sum }}{n}_i}{\displaystyle \underset{j=1}{\overset{k}{\sum }}{n}_j}$.

To do so, multiply the sums and show that for all pairs i, j, the coefficient of the term $n_i{n}_j$ is greater in the expression on the left than in the one on the right.