Solution
The Only Child in a Poisson World
Show the problem again
In a large population the number of children $K$ in a family is Poisson with mean 2 (childless families are allowed). A child is picked uniformly at random from all children in the population. What is the probability that this child is an only child (has no siblings)? Give an exact expression; a decimal rounded to 3 or 4 places is also accepted.
Worked solution
Choosing a child uniformly is choosing a family with probability proportional to its size $K$. The size-biased distribution is $P^*(K=k) = k\,P(K=k)/E[K]$. For $k=1$: $P^*(K=1) = 1\cdot 2e^{-2} / 2 = e^{-2}$. In general $P^*(K=k) = e^{-2}2^{k-1}/(k-1)!$, so $K-1$ is again Poisson(2): a random child has on average 2 siblings, not 1. The answer is **e^(-2)** $\approx 0.1353$. By contrast, a randomly chosen family with at least one child is a one-child family with probability $2e^{-2}/(1-e^{-2}) \approx 0.313$, more than twice as often: families are counted once, children are counted once each.
Source: Size-biased sampling (the family-size / friendship paradox), standard in texts on length-biased sampling. Statement written for AxiomIQ.