The king comes from a family of 2 children. What is the probability that the other child is his sister?
I acquired the correct answer $\frac{2}{3}$ which is boggeling my mind. Under any given circumstances prior to acknowledging this I would've naively, with great confidence, claimed that the correct answer evidently ought to be $\frac{1}{2}$. Is the result $\frac{2}{3}$ due to the fact that the other child already exists? Would it be correct to assess that had the other child not yet been born then the probability of it being female would've been $\frac{1}{2}$?
This is mostly right. There is also the issue of "ordering" of the two children; for simplicity, say, "first child" and "second child."
The problem (with answer $2/3$) as stated is asking for the probability that the other child is female, given that there are two children and one is male.
The other problem (with answer $1/2$) can be stated as asking for the probability that the second child is female, given that the first child is male.
If you consider the four outcomes MM, MF, FM, FF, you see that the first problem restricts the outcome space to MM, MF, FM, while the second problem restricts the outcome space to MM, MF.