The Only Child in a Poisson World

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.

• bias
• distributions

Source: Size-biased sampling (the family-size / friendship paradox), standard in texts on length-biased sampling. Statement written for AxiomIQ.