Date: Aug 27, 2013 5:25 AM
Subject: § 324 Boys and Girls Revisited

§ 324 Boys and Girls Revisited 

In MathOverflow Steven Landburg asked: Consider a country with n families, each of which continues having children until they have a boy and then stop. In the end, there are G girls and B = n boys.
Douglas Zare's highly upvoted answer to this question computes the expected fraction of girls in the population and explains why we shouldn't expect it to equal 1/2. {{This "explanation" is grossly mistaken. Of course 1/2 is the correct answer, cp. § 302.}} My current question concerns a different statistic, namely the probability that there are more boys than girls (after all families have finished reproducing). This probability turns out to be exactly 1/2, and I'm looking for an intuitive explanation of why.

There is a very simple explanation for the lacking intuition, namely: intuition is not lacking but the answer to the Google question: In a country in which people only want boys concerning the expectation E[G/(G+B)] of G/(G+B) is absolutely uncorrelated to the expected answer.
This would have become obvious if the original formulation (the ratio of boys to girls) had been taken literally by calculating, instead of E[G/(G+B)], the expectation of B/G which is E[B/G] = oo. From this result certainly nobody would have concluded that the official answer is false. [Hilbert7Problem]

... it appears to be the exact opposite of the truth. If the question is "What is B/G?", and if the official answer is "1/2" {{for B/G it is 1}}, and if the correct answer is "a random variable with expected value oo" , then recognizing the correct answer would lead not nobody, but everybody, to conclude that the official answer is false. [Steven Landsburg]

No, the expactation of fractions E(B/G) is not the fraction of expectations B/G = E(B)/E(G)! The expectation of the fractions E(B/G) is infinite, since some families have one boy as the first child and no girl. They contribute 1/0. But this does *not* influence "the expected fraction of girls in the population".

I can't believe that nobody in the self proclaimed "elite forum" MathOverflow opposes to Landsburg's utterings. But I find this is an extremely instructive parallel to Cantorism. Nothing could show better than this detail what an Overflow of Madness triumphs in Matheology.

Regards, WM