The author apparently doesn't understand where the confusion in the original question (or variations) comes from.
One way of phrasing it is asking,
"If you have an unrelated man and woman and they both have two children (one of which is a boy), where the oldest son of the man is a boy - what are the odds that they both have two boys?"
(There are any number of variations on this - what are the odds of the man having two boys? what are the odds of the woman having two boys what are the odds of the man but not the woman having two boys .. all the same problem, but stated differently).
This 'loads' the question by implying (superficially incorrect) that there might be a difference between the chances of a man and a woman having two boys.
Next up is a bit of probability theory. In the case of the woman, no order is stated, so the chances of her two children have no connection - the events are unrelated. The man, however, has as a first child a boy (which eliminates the possibility of this being a girl).
And as for the overall birthrate of men vs women or the possibilities of having twins/triplets/etc (and their male/female ratio) ... well, that's really out of the scope of a fairly trivial question statement such as this.
No, you're wrong. The thing is that there are two kinds of confusion that arise from Atwood's problem. The first kind of confusion comes from not understanding how probabilities work, which you discuss. The second kind -- which is what Paul Buchheit is talking about -- comes from noticing that the statement "I have two children, and one of them is a girl" can be parsed in two different ways. It can be parsed as, "I have two children, and at least one of them is a girl," or as "I have two children, and the gender of one of them is #{my_first_child.gender}." Despite what many commenters in this thread are saying, these are not the same.
That's the real problem here: the English language is inexact. The words that Atwood used to describe the scenario actually describe at least two mathematically distinct scenarios.
The Monty Hall problem suffers from this fact, too, but not as badly -- because both interpretations yield the same conclusion, namely, that you should switch doors. It's just that under one interpretation, you get a car 1/2 of the time, and under the other you get it 2/3 of the time. Also, under the interpretation that yields a car 1/2 the time, it's logically implied that the host is willing to open a door and reveal a car -- which most people use to rule out that interpretation, if only subconsciously.
Maybe this is because English is my first language - but I cannot imagine how "I have two children, and one of them is a girl" could mean: "I have two children, and the gender of one of them is #{my_first_child.gender}." - if someone was to confer that meaning I would expect him to say: "I have two children, and the gender of my first child is girl". But frankly as someone already pointed out - normally the sentence "I have two children, and one of them is a girl" would implicitely mean that the other child is a boy.
It's the difference between the situation where a child is chosen and then the gender announced, and the situation where a girl is found, and then the gender announced. If you started out saying "Is one of them a girl?", and only continuing if the answer is yes, then the probability changes.
Actually, even under the "correct" interpretation of the host's actions in the Monty Hall problem (where he knows where the car is and will never open that door and the participant knows this), there's still an analogy, an even closer one, to this ambiguity.
Just as here, if you ask the question in a certain way, the choice of the person, when they have the choice, becomes important (which gender to announce in case of GB/BG), in Monty Hall, if you ask the question in one common way of asking it, the choice of the host becomes important. The host can choose which door to open if they're both goats. If, in that case, the host will pick randomly, the probability of a win by switching remains 2/3. But suppose you picked door 1, and the host will always prefer door 3 when he can, for whatever reason, and you know this. Then given the information that he opened 2 or 3, the probability to win is 100%/50%, respectively. So if you ask the Monty Hall question this way: "I picked a door and the host opened another, what's my probability of winning now?", to get 2/3 the question should include the information on the host's random selection between 2/3 when the car is in 1. Admittedly that's a bit pedantic, but there you go.
"I have two children, and the gender of one of them is #{my_first_child.gender}." Are they not the same because of FIRST_child (which would be true), or is the FIRST just an accident here? I for one don't see how the original statement could be interpreted as describing the second algorithm. We clearly get the information that there is a girl among the children, and nothing is said about it being the first child.
Basically we have a situation where we see a person and we know he has two children, at least one of them being a girl. If it is impossible to derive correct probabilities from that information, then probability theory is useless. There is no algorithm, it is just a situation.
In retrospect using "#{my_first_child.gender}" was a pretty big mistake. I should have put "#{one_of_my_children.gender}". For some reason I thought the former would actually be less confusing than the latter.
One way of phrasing it is asking,
"If you have an unrelated man and woman and they both have two children (one of which is a boy), where the oldest son of the man is a boy - what are the odds that they both have two boys?"
(There are any number of variations on this - what are the odds of the man having two boys? what are the odds of the woman having two boys what are the odds of the man but not the woman having two boys .. all the same problem, but stated differently).
This 'loads' the question by implying (superficially incorrect) that there might be a difference between the chances of a man and a woman having two boys.
Next up is a bit of probability theory. In the case of the woman, no order is stated, so the chances of her two children have no connection - the events are unrelated. The man, however, has as a first child a boy (which eliminates the possibility of this being a girl).
This is another variation on the Monty Hall problem, http://en.wikipedia.org/wiki/Monty_hall_problem
Read up on some historical background on this here, http://www.marilynvossavant.com/articles/gameshow.html
And as for the overall birthrate of men vs women or the possibilities of having twins/triplets/etc (and their male/female ratio) ... well, that's really out of the scope of a fairly trivial question statement such as this.
[edit for non-trivial detail]