There are four equally likely combinations (under the [both false!] assumptions of equal and independent sexes for children in the same family): MM, FM, MF, and FF; if you know that there is at least one male (or at least one female) you eliminate one of those possibilities, leaving the relative probabilities of the other three still equal.
So, knowing no additional information, the chance of one male and one female is two-fourths, or one-half.
Knowing that there is at least one male (eliminating FF), or at least one female (eliminating MM), the probability of one male and one female is 2/3.
If you know the sex and birth order of one, you eliminate two possibilities, retaining the relative probabilities of the remaining ones as equal, so if you know the first is male, eliminating FM and FF, then the probability of one male and one female is 1/2 (and similarly, mutatis mutandis, with other sex and birth order combinations, which produce the same result eliminating different pairs of possibilities.)
> Knowing that there is at least one male (eliminating FF), or at least one female (eliminating MM), the probability of one male and one female is 2/3.
Don't you always know that there is at least one male or one female?
I mean, if A="there is at least one male" and B="there is at least one female" you're telling me that if you know that A holds the probability is 2/3 and if you know that B holds the probability is 2/3.
But, knowing no additional information, you KNOW that A and/or B holds!
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.
> Don't you always know that there is at least one male or one female?
Knowing that there is at least one male or at least one female eliminates zero possibilities.
Knowing that there is at least one male or knowing that there is at least one female eliminates one possibility (a different one for each case, but the difference is immaterial to the probability of a mixed pair).
> 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.
1/2
And if you know you will be told the sex of one child, with equal probability as to which the probability remains 1/2 when you are told, even though knowing without that constraint on how you will know makes it 1/3.
Because then the possibilities are (assume you are told “male”)
>> 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.
> 1/2
So you say it's 1/2 even though you know that as soon as you read the message you will update it to 2/3. Is that right?
The message says that (at least) one of them is a girl or that (at least) one of them is a boy. In either case, you state that the correct probability is 2/3.
> So you say it's 1/2 even though you know that as soon as you read the message you will update it to 2/3. Is that right?
Nope, its 1/2 afterwards, too.
Because you knew you were going to get one, and that is additional information; for much the same reason as the scenario outlined after that in GGP, it is twice as likely that the one you would get whichever one you do get if it was not a mixed pair.
If I say that I have two children and (at least) one is a boy you will say that the probability that I have one boy and one girl is 2/3.
If I say that I have two children and (at least) one is a girl you will say that the probability that I have one boy and one girl is 2/3.
If I say that I have two children and (at least) one is a [unintelligible] will you say that the probability that I have one boy and one girl is 1/2 or 2/3?
Edit: If you're going to say that [unintelligible] could be anything other than boy or girl - and are not willing to assume that it has to be one or the other consider the following alternatives:
If I say that I have two children and (at least) one is a [word in a foreign language that you know that means girl or boy but you don't remember which one] will you say that the probability that I have one boy and one girl is 1/2 or 2/3?
If I say that I have two children and (at least) one is [of the same sex as the first child of some other person that you don't know] will you say that the probability that I have one boy and one girl is 1/2 or 2/3?
If I tell you that one kid is male, you think that the probability that there is one male and one female is 2/3.
If I tell you that one kid is female, you think that the probability that there is one male and one female is 2/3. (Right?)
If I don't tell you anything - beyond the fact that I have two kids - what's the probability that there is one male and one female?