Agreed, but then you have to account for the fact that (boy, girl) has twice the probability of occuring as (boy, boy) or (girl, girl). (But the same probability as these combined -- which matches intuition, that two given children have a 50% probability of being the same gender.) Listing the pairs as four distinct cases with equal probability makes this clear.
As I see it, the moment you determine one of the samples, X, to be a fixed member of the total population, selecting (X, Y) from the set of pairs in the population degenerates to selecting Y from the original population. (Or rather, from the original population minus X, which may or may not be significant depending on population size.)
In a general (idealized) population, the pairs (boy, boy), (boy, girl), (girl, boy), (girl, girl) are all equally likely, 1/4. Agreed? When you remove (girl, girl), the remaining combinations are 1/3 each. If you first count the pairs where the older child is a boy, you need to remove these pairs from the population before counting the pairs where the younger is a boy, or you'll count the same pair twice.
The underlying premise is that the pair of children is selected from a global set of pairs of children where distribution of traits (gender, day of birth etc) is uniform. By applying the logic you're describing, you're selecting some of the pairs twice, thereby skewing the distribution.
The stated result in the article only applies when the children are selected as a pair from the overall population. We normally don't do that, so our intuition isn't geared for it.
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Agreed, but then you have to account for the fact that (boy, girl) has twice the probability of occuring as (boy, boy) or (girl, girl). (But the same probability as these combined -- which matches intuition, that two given children have a 50% probability of being the same gender.) Listing the pairs as four distinct cases with equal probability makes this clear.
As I see it, the moment you determine one of the samples, X, to be a fixed member of the total population, selecting (X, Y) from the set of pairs in the population degenerates to selecting Y from the original population. (Or rather, from the original population minus X, which may or may not be significant depending on population size.)
In a general (idealized) population, the pairs (boy, boy), (boy, girl), (girl, boy), (girl, girl) are all equally likely, 1/4. Agreed? When you remove (girl, girl), the remaining combinations are 1/3 each. If you first count the pairs where the older child is a boy, you need to remove these pairs from the population before counting the pairs where the younger is a boy, or you'll count the same pair twice.
The underlying premise is that the pair of children is selected from a global set of pairs of children where distribution of traits (gender, day of birth etc) is uniform. By applying the logic you're describing, you're selecting some of the pairs twice, thereby skewing the distribution. The stated result in the article only applies when the children are selected as a pair from the overall population. We normally don't do that, so our intuition isn't geared for it.
http://article.gmane.org/gmane.linux.kernel/650739