Solution Found!
Consider a school community of m families, with of ni them
Chapter 3, Problem 3TE(choose chapter or problem)
Consider a school community of m families, with \(n_{i}\) of them having i children, \(i=1, \ldots, k, \sum_{i=1}^{k} n_{i}=m \text {. }\) Consider the following two methods for choosing a child:
1. Choose one of the m families at random and then randomly choose a child from that family.
2. Choose one of the \(\sum_{i=1}^{k} i n_{i}\) 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
\(\sum_{i=1}^{k} i n_{i} \sum_{j=1}^{k} \frac{n_{j}}{j} \geq \sum_{i=1}^{k} n_{i} \sum_{j=1}^{k} n_{j}\)
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.
Questions & Answers
QUESTION:
Consider a school community of m families, with \(n_{i}\) of them having i children, \(i=1, \ldots, k, \sum_{i=1}^{k} n_{i}=m \text {. }\) Consider the following two methods for choosing a child:
1. Choose one of the m families at random and then randomly choose a child from that family.
2. Choose one of the \(\sum_{i=1}^{k} i n_{i}\) 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
\(\sum_{i=1}^{k} i n_{i} \sum_{j=1}^{k} \frac{n_{j}}{j} \geq \sum_{i=1}^{k} n_{i} \sum_{j=1}^{k} n_{j}\)
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.
ANSWER:Step 1 of 2
(a)
We are asked to show that method 1 is more likely than method 2 to result in the choice of a firstborn child.
Let \(n_{1}\) denote the number of families with 1 child.
Let \(n_{2}\) denote the number of families with 2 child.
Let \(n_{k}\) denote the number of families with k child.
We are asked to compare the two methods \(M_{1}\) and \(M_{2}\) for selecting children.
\(\begin{array}{l}M_{1}= \text{ Picking one of the } m \text{ families and we choose random children from that}\\
M_{2}=\sum_{i=1}^{k} i n_{i}\end{array}\)
Let E be the event which define the first born child is randomly chosen.