So I ran accross the following problem today on a forum:
"In an attempt to reduce male birth rates, feminists have passed a law which forces families to stop having children after their first male child. After this law is passed, what is the expected ratio of male children to female children?"
It's unclear whether families are supposed to continue having children until their first male child or if they may stop at any earlier point, but let's assume the former. Let's also assume that everybody obeys the law, and that either sex has an equal chance of being born.
I thought I had a simple solution, because the problem is logically equivalent to the following process:
- Generate N infinite random sequences of boys and girls
- Cut each sequence after the first occurrence of a boy
- Concatenate the results and measure the ratio of boys to girls
This is equivalent to generating a random sequence of boys and girls by stopping at each boy but then continuing again (up to N times), so the only difference from a normal random sequence is that this sequence always ends with a boy. However, if N grows to infinity, it seems like this last element (which can never be reached) becomes irrelevant, so the ratio of boys and girls should be 1:1 like in a normal infinite random sequence. This seems perfectly logical to me, but a few people kept insisting that it is wrong, ranting about "biased estimators" and claiming that the real ratio will be biased in favor of females.
Is my reasoning flawed? If so, why?
Contrary to some suggestions, I don't think this question is a duplicate. It asks about the validity of a particular approach to solving the problem, not just for a solution.