1/3 would be correct if you set aside the issues about 'given' information addressed in the post. And I think the post has a valid point, that we have to be careful to appropriately understand biases and uncertainties in our data sources. But your reading of it, I think, demonstrates the real weakness of this part of the OP's discussion, namely that he's started with, and trying to expand from, a basic, 'classic' problem that people notoriously have difficulty agreeing on (e.g. http://en.wikipedia.org/wiki/Maryln_vos_Savant#.22Two_boys.2...). So as a pedagogical exercise, or as an attempt to get his point across, he has to trip over all the misunderstandings that cause people to think in the simple, classic case, that the answer is 1/2.
I think he would have been better off expanding the classic coin example, only changing the story such that a naive research assistant is tasked with flipping the coin n times and bringing us the results. Then we could start by finding the probability that the coin is biased taking the research assistant at his word, and then we could reason about the probability that the research assistant either doctored the sequence when he felt they weren't 'random' enough, or the probability that he just pulled an 'HTTH' sequence out of his head without ever touching a coin.
I'll accept that I'm wrong and that the answer is 1/3 and that I just don't understand it and that I'm supporting the point that "no one understands probability, especially me", but there has been little explanation why the birth order is even a factor. Even that Wikipedia entry labels the four possibilities with "older child" and "younger child", which seems to be extra, unneeded information when the term "at least" is used to describe the number of boys independent of order.
Birth order per se doesn't matter; probability mass does. Birth order is an easy way to show that you're twice as likely to get a boy and a girl as you are to get two boys. But you can ignore birth order and say instead "let k be a binomially distributed random variable with n=2 and p=1/2. If you know k>=1 what is p(k=2)?". Then the answer is a straight comparison of p(k=2) to p(k=1), where we note that '2 choose 1' is 2, and '2 choose 2' is 1. Note that ordering per se isn't entering into this, but to someone without probability background, it's a bit obtuse. Describing the probability space in terms of birth order is not strictly necessary but helps a lot of people grasp the concept.
i think it's like this....
interpretation 1.. we select a random FAMILY from the set of all families with two children.one of the children happens to be a boy. {BB, BG, GB} are possibilities. P(BB) = 1/3
interpretation 2.. we select a random CHILD from the set of all families with two children. one of the children happens to be a boy. in this case we have the following possible combinations [B1B2, G1B3, B4G2]. we know it's a boy, so what's the probability that B1 or B2 was selected? 1/4+1/4 = 1/2.
I think he would have been better off expanding the classic coin example, only changing the story such that a naive research assistant is tasked with flipping the coin n times and bringing us the results. Then we could start by finding the probability that the coin is biased taking the research assistant at his word, and then we could reason about the probability that the research assistant either doctored the sequence when he felt they weren't 'random' enough, or the probability that he just pulled an 'HTTH' sequence out of his head without ever touching a coin.