The distant country of Toomalia has a birthrate of 67 boys for every 33 girls. In an effort to restore a 50:50 gender ratio in the country, Toomalian policymakers institute a mandatory new social policy: Couples seeking to have children may continue to do so as long as they bear only girls, but must stop as soon as they bear one boy.
Assuming the Toomalian people i) rigorously adhere to the policy, and ii) do not selectively abort their offspring, what will the net effect on the boy:girl birth ratio be when the new policy is put into effect?
Note that you may assume the gender of each conceived child is independent from the gender(s) of its siblings, but you should not assume that every couple wishes to conceive as many children as possible. Many couples will voluntarily stop having children after having 2 or 3, for example.
Answer
The ratio will stay the same. I'm not even sure how to prove that, since it seems so self-evident.
One simple approach:
Since the gender of a child is independent from their siblings, we can ignore who the parents are and ask under this procedure, what will the gender of the next child born to any couple in the country be? Well it'll still have a 67% chance of boy, 33% chance of girl. And since the parents/siblings don't matter in that number, those chances will be constant, so over time as more children continue to be born, it will be in the same ratio.
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