[mofeien]

The statement "at least one of them is a boy" (<=> "I don't have two daughters") is a little more subtle than "I tell you the gender of one of them" since the former excludes one out of four possibilities (FF, thus letting us update our belief on the single-sex question to a third) while the latter implies fixing the gender of a specific one of the children (without specifying which one, and in either case the probability of the other being M is still a half, thus not giving us information towards the single sex question).

So if you tell me the gender of a specific one of them, say the youngest, then I haven't learned anything that makes my subjective probability go down that the other is the same gender.

I think in real life you will come across the second kind of statement (e. g. "my oldest is a girl") than the first kind (e. g. "I do not have two boys")

But it does not feel too weird to me that "at least one of them is a girl" will reduce the probability of the pair being single-sex to a third. In fact if you further tell me that both "an least one of them is a girl" and "at least one of them is a boy", the probability of the pair being single-sex will go to zero and this seems perfectly reasonable

[kgwgk]

Do you agree with the following?

> I tell you I have two children and that (at least) one of them is a boy, and ask you what you think is the probability that the pair is single-sex.

1/3

> I tell you I have two children and that (at least) one of them is a girl, and ask you what you think is the probability that the pair is single-sex.

1/3

If you don’t, why not?

If you do, what’s your answer to the following question?

> I tell you I have two children and that I’ve just sent you an email with the sex of (at least) one of them, and ask you what you think is the probability that the pair is single-sex.

Will your answer change after you have a chance to check your messages?

[kqr]

In this scenario, you are subtly changing the meaning of "single-sex".

In the first two cases, "single-sex" means "the same specific sex as the child you know the sex of" whereas in the last case it means "the same sex as a child that can still have two possible sexes".

If you would say,

> I tell you I have two children and that I’ve just sent you an email with the sex of (at least) one of them, and ask you what you think is the probability that the pair are both girls?

and then follow up with another question,

> what you think is the probability that the pair are both boys?

and then add the two probabilities up equally weighted, you might see why 1/2 is the reasonable answer in that case.

(And why opening up the email in question would reduce the probability of one of the questions to 0, and the other to 1/3.)

[kgwgk]

As you find “both girls or both boys” problematic for some reason maybe we can discuss the following questions instead - where hopefully there is no subtle change of meaning.

————-

Do you agree with the following?

> I tell you I have two children and that (at least) one of them is a boy, and ask you what you think is the probability that I have one boy and one girl.

2/3

> I tell you I have two children and that (at least) one of them is a girl, and ask you what you think is the probability that I have one boy and one girl.

2/3

If you don’t, why not?

If you do, what’s your answer to the following question?

> I tell you I have two children and that I’ve just sent you an email with the sex of (at least) one of them, and ask you what you think is the probability that I have one boy and one girl.

[kgwgk]

I’ll come back to your reply later, but I would appreciate it if you could give a precise answer to the questions I asked.

It may help to find a common understanding on top of which we can build a clear discussion of the subtleties involved.