Children and grandchildren in a random family tree

Question

A founder has a random number of children $N$ , where $N$ is equally likely to be $0, 1, 2,$ or $3$ . Each child independently has a random number of children $M$ , where $M$ is equally likely to be $0, 1,$ or $2$ .

What is the expected number of grandchildren?

Accepted Answer

The number of grandchildren is a sum: add up each child's number of children. But the number of children N is itself random, so this is a random-length sum. Condition on N and use the tower rule. Step 1, the per-child mean. Each child has, on average, E[M] = \frac{0 + 1 + 2}{3} = 1 children of their own. Step 2, freeze the number of children. Given N = n, the grandchildren count is a sum of n independent terms each averaging 1, so E[	ext{grandchildren} \mid N = n] = 1 	imes n = n. Step 3, average over N. The founder has, on average, E[N] = \frac{0 + 1 + 2 + 3}{4} = 1.5 children. So E[	ext{…

Children and grandchildren in a random family tree

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