A King's Decree
King Leo of the kingdom Miley dotes on his two daughters so
much that he decides the kingdom would be better off with more
girls than boys. He makes the following decree: All
child-bearing couples must continue to bear children until they
have a daughter!
To avoid overpopulation, he makes an additional decree: All child-bearing
couples will stop having children once they have a daughter! His subjects
immediately begin following his orders.
After many years, what’s the expected ratio of girls to boys in Miley? Keep in
mind that the likelihood of each baby born being a girl is, of course, 50 percent.
Solution
Don’t overthink this. Each baby born is as equally likely to be a boy as a girl. Therefore, the ratio of girls to boys must be 1:1. It’s as simple as that—honestly.
You might be tempted to solve this problem in a more complicated way. Suppose
there are N child-bearing couples. Half of them will have only 1 child: a girl.
Half of the other half (or N/4) of them will have two children: one boy and one
girl. Half of the remaining quarter (or N/8) will have three children: two boys
and one girl. And so on…
In this generation, the total number of children can be found from an infinite
geometric series:
N + N/2 + N/4 + N/8 + N/16 + …
The sum of this series is 2N. Since there will be exactly N girls (1 per
couple), girls are 50 percent of this generation!
Now, you might point out that it’s impossible for families to have, say, 20 or
more children, in the event that they keep having boys. (And, if not impossible,
certainly undesirable!) The calculation above might change if King Leo allows
couples to keep their family sizes to a more manageable level.
But even with those restrictions that may lead to more complicated math, the
conclusion will always be the same: The ratio is 1:1 as long as each baby born
has an equal chance of being a girl or boy.
(From Popular Mechanics)