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PhilGoetz comments on Anticipating critical transitions - Less Wrong
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He did answer the question he posed, which was "What is the expected fraction of girls in a population [of N families]?" It's not an unmeasurably-small hair. It depends on N. When N=4, the expected fraction is about .46. If you don't believe it, do the simulation. I did.
I believe the mathematics. He is correct that E(G/(G+B)) < 0.5. But a "country" of four families? A country, not otherwise specified, has millions of families, and if that is interpreted mathematically as asking for the limit of infinite N, then E(G/(G+B)) tends to the limit of 0.5.
To make the point that this puzzle is intended to make, about expectation not commuting with ratios, it should be posed of a single family, where E(G)/E(G+B) = 0.5, E(G/(G+B)) = 1-log(2).
But as I said earlier, how is this puzzle relevant to the rest of your post? The mathematics and the simulation agree.
Estimating the mean and variance of the Cauchy distribution by simulation makes an entertaining exercise.
Thinking about betting $15,000 on a math problem, to be adjudicated by the outcome of a computer simulation, made me wonder how we know when a computer simulation would give the right answer. Showing the results for the similar-looking but divergent series is the simplest example I could think of of when a computer simulation gives a very misleading estimate of expected value, which is the problem this post is about.
The question asked about a country. Unless you're counting hypothetical micro-seasteads as countries, the ratio is within noise of 50%.