Flim-Flam Man

Prof. Steve Landsburg, AKA "The Burg" has a response of sorts to criticism of his "percentage of girls" puzzle from yours truly and Lubos Motl. That response was to challenge Lubos (I am the blogger-who-may-not-be-named) and all comers to a \$15,000 sucker bet. Lubosh has called the bet a "borderline cheat" but I don't see any border at all.

The problem is that the subject of his bet is quite different from the original puzzle and his answer to his new problem is different than his original answer - and, not incidentally, now correct in a certain approximation.

Here is the original problem:

There’s a certain country where everybody wants to have a son. Therefore each couple keeps having children until they have a boy; then they stop. What fraction of the population is female?

Well, of course, you can’t know for sure, because, by some extraordinary coincidence, the last 100,000 families in a row might have gotten boys on the first try. But in expectation, what fraction of the population is female? In other words, if there were many such countries, what fraction would you expect to observe on average?...

It seems to me that Landsburg is pretty clearly indicating that he is talking about a country with plausible country sized reproducing population, e.g. comparable or larger than 100,000. When he did his calculation, though, he initially did it for one family, and claimed that that was a perfectly good surrogate for a country - though he has since apprently changed his tune on that. The problem with that, as Doug Zare and others have pointed out, is that one family is a biased estimator for the population.

Let's take a look at his "new" problem - I call it new because it sure doesn't look like the original to me.

I specified that the answer is to be interpreted in expectation, since the actual fraction of girls could be anything at all due to statistical flukes.

I say the answer depends on the number of families in the country, but in no case is it 50%. Lubos insists that the correct answer is 50%.

Now the best way to settle such a dispute is to go to the mathematics. But since Lubos seems unable to follow the mathematics, the next best way is to run a simulation. So I propose the following terms: We’ll randomly choose five graduate students in computer science from among the top ten American university departments of computer science and have them write simulations for a country starting with, say, four couples, each having one child per year and stopping when they have a boy. We’ll let this run for a simulated 30 years and then compute the fraction of girls in the population.

[Edited to add: If Lubos (or anyone else) prefers to run the simulation till every family is complete (as opposed to a fixed number of years), that's fine with me.

I think this is a ridiculous way to frame a bet, and I doubt that Lubos will bite. If you want to do a simulation, specify the code and let us decide if you capture the problem, not some anonymous grad students. I don't have that kind of cash to play with, and running an internet gambling operation is probably illegal anyway, but let me suggest a more natural formulation of the problem: start with 100,000 families, the number mentioned in your first post.