Assume each child is independently equally likely to be a girl or a boy. Every family continues having children until its first boy, then stops. For a population with many independent families, derive the expected number of girls and boys per family and the limiting ratio of girls to boys. State why the random family size does not create a different population ratio. ### Constraints & Assumptions - Ignore demographic differences other than the stopping rule. - Each birth is independent with probability `1/2` for either outcome. - Interpret the population ratio through totals across an increasing number of families. ### Clarifying Questions to Ask - Are births independent and identically distributed? - Are we asked for a per-family expectation or the ratio of aggregate counts? - Does every family continue until a boy is born? ```hint Write the family outcome distribution A family with exactly `g` girls has sequence `G^g B`. ``` ### What a Strong Answer Covers - The geometric distribution and almost-sure stopping. - Expected girls and boys per family. - A law-of-large-numbers argument for aggregate counts and the distinction between a ratio of expectations and an expectation of a ratio. ### Follow-up Questions - What changes if a boy is born with probability `p`? - What if families stop after at most three children? - Why is selecting a random child different from selecting a random family?