I think I've spotted where your misunderstanding is.
BG is only not the same as GB if there is some other information available - which was born first, what their names are, hair colour, etc., because then you'd be saying something like
Boy born first, Girl born second
Girl born first, Boy born second
and those are two distinct possibilities. The point is that they are only distinct if you have this extra information, which we don't. We have no way of differentiating between the two children except for gender, and therefore BG and GB both just say 'one male child and one female child'. You might assume that the order of the two letters specifies the order in which the children were born, but that's a false assumption because it's not stated anywhere.
If you include the order, there are four possibilities: a) Boy first, Boy second, b) Boy first, Girl second, c) Girl first, Boy second, d) Girl first, Girl second. If you don't, there are three possibilities: a) two boys, b) two girls, c) one boy and one girl. The unspecified information about order is the subtle but important difference.
No, they are correct in saying that GB and BG are distinct, even with no other information.
It is not order that is important, but considering each child as a distinct entity.
The 50% chance of being a boy and 50% chance of being a girl applies to a single independent child. When enumerating the possible combinations we need to first enumerate the possibilities for each child, and then combine these two enumerations into our overall enumeration.
To help us distinguish between the two children, let's call one Sam and the other Alex. There's no ordering over them, it's just to help us tell which one we're talking about.
Sam can be: Boy, Girl
Alex can be: Boy, Girl
Combining those gives us:
Sam is a Boy, Alex is a Boy
Sam is a Boy, Alex is a Girl
Sam is a Girl, Alex is a Boy
Sam is a Girl, Alex is a Girl
Or, to use a shorter notation, BB, BG, GB, GG.
Using oldest and youngest is just another handy way of distinguishing between the two children. Even if they were nameless, faceless children with no distinguishing characteristics other than gender we still need to consider then separately, each as their own entity with their own enumerations of possible genders.
Where the parent commenter is wrong is saying that (correctly) considering GB and BG as distinct combinations is the same as considering "Oldest Boy born on Tuesday and Youngest Boy born on Tuesday" distinct from "Youngest Boy born on Tuesday and Oldest Boy born on Tuesday". They are not. When you stop thinking about ordering and instead think about the enumerated states of each entity (child) involved it becomes clear both are saying the same information. They are not distinct, but the same combination phrased differently.
The parent commenter is basically saying BB should be distinct from BB just because the first one talks about Sam first and the second one talks about Alex first. This is clearly wrong.
Hmmm I think we're saying more or less the same thing. The point I wanted to make is that
"I have a son and a daughter"
is the same as
"I have a daughter and a son"
unless you qualify the statement with further information - names (Alex and Sam from your post) would be an example of that further information. My feeling was that ars is implicitly 'filled in the blanks' somewhere, treating the two children as distinct when in fact they have to be interchangeable for the purposes of the original (13/27) calculation.
It's the lack (or not) of such details that alters how the probability is calculated.
I admit though that I'm still chasing myself in circles trying to understand the whole thing so take this reply with a pinch of salt ;-)
BG is only not the same as GB if there is some other information available - which was born first, what their names are, hair colour, etc., because then you'd be saying something like
Boy born first, Girl born second
Girl born first, Boy born second
and those are two distinct possibilities. The point is that they are only distinct if you have this extra information, which we don't. We have no way of differentiating between the two children except for gender, and therefore BG and GB both just say 'one male child and one female child'. You might assume that the order of the two letters specifies the order in which the children were born, but that's a false assumption because it's not stated anywhere.
If you include the order, there are four possibilities: a) Boy first, Boy second, b) Boy first, Girl second, c) Girl first, Boy second, d) Girl first, Girl second. If you don't, there are three possibilities: a) two boys, b) two girls, c) one boy and one girl. The unspecified information about order is the subtle but important difference.