Probability Problem: Find P(Product=6 | Die1=2)

[haruspex]

If you just report what you saw then one can make a good case that the probability the other child is a girl is 1/3 (because if you see a girl it eliminates the case BB, leaving three equally-likely cases GB, BG, GG

To be clear, this is the formulation I am challenging: if you see Albert with a daughter, and know by some means there is another child, that other child is equally likely boy or girl, not 2:1 a boy.

No selection bias was used in getting the 1/3

Yes there was. If you had seen a boy and the other child was a girl, you would not have reported that at least one was a girl. In the standard (valid) formulation you will report that at least one is a girl whenever that is true. In the above formulation you will not achieve that.

So, if there are at least two children there are four possibilities: if at least one is a girl you know BB is not an option

Yes, but it's not that simple. In order to deduce the 1/3 answer you have to make the assumption that if either is a girl, you would have the information that at least one is a girl. This is where Ray's argument above breaks down. Having seen a girl, he knows at least one is a girl, but there is a case where at least one is a girl yet he does not have that information.

you are missing the point: introductory material is introductory material because the point is to stress the basic ideas: complicating factors are introduced later.

That seems to me a total misrepresentation of the matter.
There is much evidence that this problem is rather subtle. It was not properly understood by some HSC examiners, and appears to have fooled the eminent Ray Vickson. (You have not posted a view on Ray's argument, so I cannot tell whether it fooled you too. )
To make it 'introductory level', it will be essential to minimise confusion. The advanced level could then look into the epistemological niceties.

I view your argument as "this problem is always poorly worded in tests, so we have to teach students to interpret it in a certain way". Well, that's nothing to do with teaching mathematics; that's teaching interpretation of sacred texts.

 

[haruspex]

"If you tell your partner that at least one of them is a little girl you are misrepresenting the information that you have ..."
That is blatantly false. If you see a girl you know that there at least one - what else could it be, none? What you don't have is evidence there is exactly one.

First, perhaps Orodruin should have written that you are understating the information that you have, i.e. you have thrown some information away. On that he is correct, and is likely to be the case whenever someone fails to state how they know the information.
Secondly, your argument supports my case exactly! If you believe that relaying the information as "at least one of them is a little girl" correctly conveys the information that you have (that you saw a girl and have no idea of the gender of the other child) then you are forced to the view that the statement "at least one of them is a little girl" does not support a 2:1 likelihood that the other is a boy.