Back to "The Burg"

Steve Landsburg brings us this little post-Channukah present:


I think he mangles the logic and presents a totally bogus answer.* Let's read it:

•Here’s the wrong answer: Every birth has a 50% chance of producing a girl. This remains the case no matter what stopping rule the parents are using. Therefore the expected number of girls is equal to the expected number of boys. So in expectation, half of all children are girls. {OK, Compare paragraph below}
•Pretty convincing, eh? So why is it wrong? Well, actually, most of it is right. Every birth has a 50% chance of producing a girl — check. This remains the case no matter what stopping rule the parents are using — check. Therefore the expected number of girls is equal to the expected number of boys — check! But it does not follow that in expectation, half of all children are girls! { Yes, Steve, it does!!}
•To see why not, let me tell you about the families who live on my block. There are 3 families with four girls each (and no boys), and one family with 12 boys (and no girls). Altogether, that makes 12 girls and 12 boys — equal numbers! On average, each family has three girls and three boys. Nevertheless, the fraction of girls in the average family is not 50%. It’s 75% (the average of 100%, 100%, 100%, and 0%).

But Prof L, that's not the question you asked. You asked what fraction of the population was female, not what fraction of the average family was female. The answer to the question you asked *is* 50% just like the fraction of female children on your block is 50%. The question you intended to ask, but didn't is: What is expected value of the fraction of the average family that is girls? The answer to that question is:

0*1/2 + (1/2)*1/4 + (2/3)*1/8+ ... =

UPDATE: Maybe I really should stop picking on Landsburg. A number of his commenters has pointed out exactly what's wrong with his logic, but he still doesn't get it. I thought he was just impossibly stubborn, but maybe he's got a brain block or something. He can't even grasp that the fraction of girls in his (hypothetical) neighborhood (12/24) is equal to 1/2. It still drives me nuts that thousands of dim students treat this guy as an authority.

Here's a nice example of the way Steve deals with a commenter who rather politely and repeatedly failed to "get" his point:

Tom: You are definitely too stupid to think about this problem.

UPDATE II: I was remiss in not linking to Doug Zare's original post at MathOverflow. Note that his answer ((1/2) - 1/4k)(a)differs from the Burg's and (b)is very close to 50% for a country with a plausible number of families k. Both his answer and Landsburg's require that you include biologically impossible cases - families with arbitrarily large numbers of children)

It's worth noting that answers to such questions depend critically on how you translate the real world into math and vice-versa. Landsburg, in my opinion, fails egregiously.

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