Sultan's law involving outnumbering

Question

A Sultan wanted to increase the number of women in his country, as compared to the number of men, so that men could have larger harems. (Sorry ladies!) To accomplish this, he proposed the following law: As soon as a mother gave birth to her first son, she would be forbidden to have any more children. In this way, the Sultan argued, some families would have several girls, and only one boy, but no family would have more than one boy. It would not be long before the females greatly outnumbered the males. Do you think the Sultan’s law would work?

This problem is considered a math problem which i think is pretty interesting. I've read this problem to a couple of people and here is what we thought:

1. What if every women gives birth to a boy, then this would not work.

I'm not sure what factors are controlled in this problem but any ideas?

Answer 1

No, it would not! the shortest explanation is the sanity check, "why does when I stop having babies influence the babies I have already had?"

Though, perhaps easier to understand is, if you look at a single woman, the expected number of male children is 1, and the expected number of female children is 1/2 for the first + 1/4 for the second + 1/8 for the third + ... =1.

Answer 2

Making reasonable assumptions about births, every child born still has an equal chance of being a boy or a girl. So the population balance would still be expected to be $50$-$50$.

(Or if for some reason it wasn't $50$-$50$ previously, it would still be expected to be the same as it was before.)

The fact that no family can have more than one boy might seem to increase the proportion of girls, however it is balanced by the fact that $50$% of families would have no girls at all.

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Of course if there is the possibility of determining the child's gender before birth and the option of terminating a pregnancy, then anything could happen. Everyone knows of societies (one in particular) in which controls on childbearing have actually been tried. But this then becomes a sociological question rather than a mathematical one.