We don't care about which came first or second, only what gender each child is.
Thus the answer, to the question given the information you have, is 50%. The only possible outcomes are girl-girl or girl-boy (where order is irrelevant.)
And this is absolutely NOT the Monty Hall Problem. I don't know why some people are making that reference. The Monty Hall Problem contains three possible choices and one is eliminated by the host, this is what makes the statistical math interesting in that problem. None of that is happening here.
Lets look a the exact wording of the question:
> a family has two children. You're told that at least one of them is a girl. What's the > probability both are girls?
We have a family with two children. Assume we don't know their gender. We'll represent them as XX.
We are told one of them is a female. So now they are represented as GX (remember GX = XG, since order doesn't matter.)
You are left with the question what is the probability that X is female? Well there are only two choices, F and M, and we are told elsewhere that the probability of having a girl is 50/50.
> Assume that the probability of having a girl or boy is 50% and that the birth order has > no effect on the probability.
So the chance of X being female is 50%. Thus the answer is 50%.
You can't say birth order doesn't matter and then use birth order to say the FM and MF are different results. The only possible results are FM and FF (since birth order is irrelevant.)
It's the fact that there are twice as many boy-girl families as there are girl-girl families in the world.
Do you mean “interpretation” or “alternative problem”.
Because if it’s an “interpretation” of the original problem you’re indeed making assumptions to fill the unspecified information.
If you mean that it’s an alternative problem which has a definite solution I agree. It’s a different problem and its relevance to the original one is to illustrate that additional assumptions were required.