Steve Landsburg gives a wrong answer, different from 50%, while CommunistStalinistSwine gives a correct one, 50%.
The chance that a birth produces a girl remains 50% regardless of the laws. So unless some children are deliberately aborted or otherwise killed, the proportion of girls remains 50%.
Another question is the percentage of girls in a family with child(ren) averaged over all families with child(ren). As CommunistStalinistSwine correctly writes, the answer is
(0/1) * 1/2 + (1/2) * 1/4 + (2/3) * 1/8 + ... =Your humble correspondent kindly added the analytic and numerical result for the sum. Have you ever encountered the Taylor expansion for "ln(1-x)"? And substituted "x=1/2" into it haha?
= 1 - ln(2) =
= 0.30685...
Note that the two results, 31% and 50%, differ. It's because each rare family with many daughters raises the family average just a little bit - because it is just one family - but it raises the proportion of girls in the society a lot - because it offers many girls.
This discrepancy is actually an example of the inability of the anthropic principle to produce quantitative predictions of any probabilities.
Imagine that the families above are Universes in a multiverse. You want to calculate the odds of being female. If you take an average Universe and an average observer in it, you obtain 31%. If you take an average observer in the whole multiverse, you obtain 50%.
It matters whether you insert additional "average something" layers into your calculation or not. In reality, such extra layers don't change the result from 31% to 50%. They may change the odds by dozens of orders of magnitude or more.
If you look for a "typical cell" of an organism, it is much more likely to be a cell of a big organism - such as a polar bear that are not endangered, just threatened - than a small one. So looking for a typical cell on the Earth ends up with a very different result than looking for a typical cell in a typical organism - which is a bacterium. The former is much more likely to be a cell of a large organism.
By the way, while solving his puzzle, Landsburg mentioned a family that has 12 boys. The parents should be arrested because according to the very laws mentioned above, the last 11 boys were born and maybe even conceived illegally.
A loophole
There is actually a subtle assumption in CIP's formula that has led to the result 31%: we only considered families that already have a son, and that have already finished their childmaking.
If we had included families that haven't had a son yet, the percentage of girls in a family averaged over families with at least 1 kid would increase and arguably stay at 50%. To calculate the percentage by a sum, however, we would have to know at least the relative proportion of 1-girl and 2-girl families at every moment. This percentage, reflecting the typical fertility of a family - in a similar way as in a society without the 1-son constraint - wasn't determined in the formulation of the problem.
Because of this loophole, you may explain the lower result of 31% obtained previously as an artifact of the selection bias - that we have only considered the "completed families" that have already received their son. This selection bias has increased the percentage of boys and decreased the percentage of girls. Without this selection bias, the average percentage of girls has to remain at 50% even in an average family under-or-after construction.
Later, I learned that this problem was asked during interviews at Google. The official answer is 50:50, thank God.
