## Friday, December 03, 2010

### Slippery Lube

This is not a real blog entry, just a page posted on December 30th, 2010, as a reply to Steve Landsburg on his blog. His entry is called "Slippery Lube". The title is very funny but that's where the quality of the article ends.

He is trying to defend a wrong solution to a simple problem about the proportion of girls in a country where the son has to be the last child in each family.

My comment is below (it's just for me and the people who Google search for "Slippery Lube") - and obviously, he is not approving comments that disagree with him:

Dear Steve,

honestly, I really don’t enjoy neverending debates and arguments about trivial issues which is why I de facto closed the thread and I surely don’t plan another blog entry about this trivial puzzle. The issue is totally clear. It’s clear that every long enough sequence of births will produce exactly 50% of girls and 50% of boys, regardless of the parents’ superstitions and their algorithms to stop reproduction at various points.

The average non-weighted proportion of girls in a family is a subtle exercise, and it’s nice that you also calculated it’s 30.6% (Taylor expansion for a 1-ln(2)), which is strictly below 50% because the small one-boy-only families are overrepresented in computing the average, but it’s not the same thing as the fraction in the society that is exactly 50% for any sustainable society or country or nation. To get the latter, one has to weight the families by their numbers in computing the average proportion, and one always gets 50% exactly in this way as long as the nation is sustainable.

The only way to get a number different from 50% is to replace the nation by a limited history that goes extinct after their families hit the wall – by having the first son. This limited suicidal sect simply can’t be called a nation or a country.

Even if you take these limited histories, the properly weighted average of the proportion of girls over such histories will still be 50%. All qualitative arguments that the number is different from 50% have errors in it. For example, if one also counts incomplete families, he could say that large families have a higher fraction of girls while the 1-child families are balanced, so the boys will never catch up. This is a wrong statement because if the couples would normally have many children, the stopping rule creates some highly-female families, but the same stopping rule also increases the proportion of families who have 1 boy and no girls – the pairs angry about the government because they wanted to have many kids but the government stopped them after the 1st kid. This demographics doesn’t exist for 1-girl-only families which means that 1-boy-only families prevail over 1-girl-only families. The full calculation, of course, implies that the two effects are balanced and the fraction is exactly 50%-50%.