Just to pile on here, there's also ambiguity around how the observed girl is selected. Consider the following framing:
I go to a random house on a random street and knock on the door. A young girl opens the door. I ask how many siblings they have and they say one. What's the probability that they have a sister?
Now it's 50% even though cosmetically it seems like it'd be fair to say that the family has at least one daughter. The reason is that once I see a girl at the door, I'm slightly more confident in that it's a GG household since a GB or BG household would sometimes show a boy opening the door (assuming the two kids are equally likely to open the door).
P(GG | G at door) = P(G at door | GG) P(GG) / P(G at door)
I go to a random house on a random street and knock on the door. A young girl opens the door. I ask how many siblings they have and they say one. What's the probability that they have a sister?
Now it's 50% even though cosmetically it seems like it'd be fair to say that the family has at least one daughter. The reason is that once I see a girl at the door, I'm slightly more confident in that it's a GG household since a GB or BG household would sometimes show a boy opening the door (assuming the two kids are equally likely to open the door).
P(GG | G at door) = P(G at door | GG) P(GG) / P(G at door)
P(G at door) = 1/2 (by symmetry)
So, P(GG | G at door) = 1 * 1/4 * 2 = 1/2