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Correct length-biased sampling from family-size survey

UpstartData Scientist · Technical Screen
EasyData ScientistTechnical ScreenStatistics & Math

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This question evaluates understanding of length-biased sampling and inference under sampling bias, specifically how sampling by individuals (children) differs from sampling by households when estimating family-size probabilities.

In a town, you visit a school and ask 100 kids: “How many children are in your family?” You observe:

  • 50 kids say their family has 1 child
  • 20 kids say their family has 2 children
  • 30 kids say their family has 3 children

Now you go to a random house in the town, knock on the door, and ask: “How many children live here?” Assume every family has at least one child, and (for this question) family sizes are only 1, 2, or 3.

Question: What is your best estimate of the probability that a randomly chosen house has exactly 1 child?

Clarify any assumptions you need about how the 100 kids were sampled (e.g., uniformly at random from all children in town).