The king comes from a family of 2 children. What is the probability that the other child is his sister?
First of all, take note about @Micah comment and suppose the male-preference primogeniture is observed.
This is a Conditional Probability question.
We'll use the followin notation:
$P(A|B)$ means the probability of event $A$ occurs given that event $B$ has occurred.
By definition on page 59 of Sheldon Ross book, we know that $ P(A|B) = \dfrac{P(AB)}{P(B)}$, where $AB = A\cap B $. Another notation is $|X|$ that means the number of elements in set $X$.
As the king comes from a family of two children, we are given two tips: (1) This family has a boy, the king. (2) The king has a sibling. What we want to know is the probability this sibling be a girl. In other words, what's the probability of the two children be each one of each gender.
Let B the event of possible children where at least one is a boy, the king. So $B=\{(b,b), (b,g), (g,b)\}$, where $(x,y)$ means the gender of each child and the possible values are $b$ for boy and $g$ for girl. Then $A$ is the event that the king's sibling is a girl, $A=\{(b,g),(g,b)\}$.
The sample space $S$ contains all possible outcomes, $S=\{(b,b),(g,b),(b,g),(g,g)\}$.
It follows that $AB=A$ and $|AB|=|A|=2$, $|B|=3$ and $|S|=4$.
So, we have:
$$P(A|B) = \dfrac{P(AB)}{P(B)}=\dfrac{\dfrac{|AB|}{|S|}}{\dfrac{|B|}{|S|}}=\dfrac{|AB|}{|B|} = \dfrac{2}{3}$$
The end.
As Martin Gardner points out regarding the similar "boy or girl" paradox (see here), a failure to specify the randomizing procedure could lead readers to interpret the question in several distinct ways. (This wording is partly Gardner's and partly the Wikipedia article author's.)
For example, the following alternative to srodriguex's interpretation also seems reasonable:
We randomly select a king. Then we are informed that the king has exactly one sibling. What is the probability that the sibling is female?
In this interpretation the answer is 1/2, not 2/3. One way to see this is by symmetry, because the fact that kings are male does not affect the calculation in this interpretation.