I'll take that bet. Write the code, but be careful to implement my second algorithm exactly.
My point is that there is nothing wrong with Jeff's question, and I believed that you made a distinction with the Monty Hall version. But your second algorithm isn't the same as Jeff's question, my mistake.
It is very clear that my argument specifically adresses the question the way Jeff has formulated it, so you can even forget the part about "your second formulation", what I meant was the question of Jeff and why there is nothing wrong with it.
In statistics, it is commonly accepted that we don't know what we don't state (for an obvious practical purpose). Therefore, when we say : "One of them is a girl", it is implied that we don't know wich one because we didn't specify it.
Therefore there is nothing wrong with Jeff's question (and the probability is 2/3), at least for anyone familiar with basic statistics and probabilities conventions (and I suspect you are), anything else is just quibble. I assumed your argument wasn't about nitpicking this kind of convention, but it turns out I was wrong. I guess nobody wins this one :)
As mentioned in another thread started by Eliezer, it's the difference between "What is the sex of one of your children?" and "Is (at least) one of your children a girl?". For the second question, the results skew to 1/3 and 2/3 because we're discarding the cases where the answer is 'no'.
The light went on for me when Paul pointed out in his update that getting a random answer for "What is the sex of one of your children?" eliminates (an unknown) one of BG and GB.
In the world of mathematical conventions you learn in school, the question is understood and you're right. In the "normal" world where you can quibble with language because there is no specific agreement, Paul is right and the question should be more precise. In both cases the conclusion remains the same : what a waste of time.