I see - I thought the first child you meet is always a boy in your example. So I am still not sure what you are calculating, but at least I understand the tuple notation :-)
I was trying to make a point that two problems that look the same actually lead to different results based on what information you are given. For example, being told someone has 2 kids and at least one is a girl then the space for this is S = {(M,F), (F,M), (F,F)}. However if I were to run into someone with two kids and saw a daughter then the information I have doesn't allow me to construct the space S above if I wanted to find the chance of say, 2 girls while accounting for age.
Instead I must consider S X {1,2} because instead of knowing that there is at least one daughter I just know that I have met one daughter whose age (whether older) is unknown to me. I know nothing of her sibling. So I am calculating the possible ways I could have met her and then the chance of 2 daughters conditioned on that space. This is a very tricky differentiation. Distinctions like this and thinking about what to condition on is what makes probability so tricky.
I must admit, I don't understand the distinction. What does age have to do with it? The information seems to be the same in both cases: a man with two kids, at least one of them a girl.
Of course thinking about age is a legit way to calculate the probability of the second kid being a girl. It is just unnecessarily complicated.
The distinction is in how the information was presented and there is a difference (let us ignore the ages and assume i met the younger one then seeing as I met one girl my space is {(F,F), (F,M)} vs {(F,F), (F,M), (M,F)} for at least one girl while accounting for age). For the second age does not matter so much as who is older and accounting for all combinations is why I present it that way. The distinction is tricky but problems like these are often covered in the conditional probabilities section of most probability theory texts.
