> I interpreted "it depends on randomness" as "speak up if you have at least one boy" or "speak up if you have at least one girl", each of which would result in 450 saying something in your example and the math works.

The point is that original problem says "I tell you I have two children and that (at least) one of them is a boy". It doesn't say "I tell you I have two children and [when you ask me to confirm whether (at least) one of them is a boy] that (at least) one of them is a boy".

Reasoning from the cases "BB", "BG", "GB", "GG" - and discarding the last one to get p(BB)=1/3 - is implicitly using the cases "BB and I tell you that at least one of them is a boy", "BG and I tell you that at least one of them is a boy", "GB and I tell you that at least one of them is a boy", "GG and I tell you something else like at least one of them is a girl".

That breaks the symmetry between the "I tell you that at least one of them is a boy" and the "I tell you that at least one of them is a girl" problems. Using the cases in the previous paragraph the answer for the former is p(BB)=1/3 but the answer to the latter is p(GG)=1.

If you want to have the same solution when you switch girl <-> boy ["one of them is a boy, what is the probability that I have two boys" <-> "one of them is a girl, what is the probability that I have two girls"] you should treat them equally.

A quite natural way to do so would be to consider the eight equiprobable cases (some of them repeated)

    BB and I tell you that at least one of them is a boy
    BB and I tell you that at least one of them is a boy
    BG and I tell you that at least one of them is a boy
    BG and I tell you that at least one of them is a girl
    GB and I tell you that at least one of them is a boy
    GB and I tell you that at least one of them is a girl
    GG and I tell you that at least one of them is a girl
    GG and I tell you that at least one of them is a girl

but then the answer to both problems is 1/2.

You can make the answer to both problems 1/3 but the eight cases that you would need to consider for that are quite unnatural:

    BB and [... discard this line somehow ...]
    BB and I tell you that at least one of them is a boy
    BG and I tell you that at least one of them is a boy
    BG and I tell you that at least one of them is a girl
    GB and I tell you that at least one of them is a boy
    GB and I tell you that at least one of them is a girl
    GG and I tell you that at least one of them is a girl
    GG and [... discard this line somehow ...]

I think at this point we agree on the math and disagree on how the original question should be read. You saw an implicit alternative to the statement as "I have two children, at least one is a girl" and assumed in the question the parent was saying exactly one of that or the original statement "I have two children, at least one is a boy", possibly chosen randomly, from among the true answers. I read it as a simple statement true statement with GG just being an undefined state for the statement generation, maybe resulting in nothing being said. We could argue about parsing English, but it seems less interesting which question was being posed if we agree on the math behind each parsing, which I think we do?

> You saw an implicit alternative to the statement as "I have two children, at least one is a girl" and assumed in the question the parent was saying exactly one of that or the original statement

I made the alternative explicit when I asked you what do you think that the probability is for two girls under the alternative.

And you said 2/3.

> I read it as a simple statement true statement with GG just being an undefined state for the statement generation, maybe resulting in nothing being said.

And with BG and GB being states that generate the statement "I have two children, at least one is a boy".

But then BG and GB cannot generate the statement "I have two children, at least one is a girl".

As a I said there are also ways to justify that the answer to both is 2/3 but they are quite convoluted.

> We could argue about parsing English, but it seems less interesting which question was being posed if we agree on the math behind each parsing, which I think we do?

Maybe we could agree that this is an undefined situation because the answer depends on how you decide to interpret the question?

More precisely, it depends on your assumptions about the (relative) value of P(you tell me that you have at least one boy|you have two boys) and P(you tell me that you have at least one boy|you have one boy and one girl).