[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.