The king comes from a family of 2 children. What is the probability that the other child is his sister? I've found this question on a book and I'd like a review in my answer.
The king comes from a family of 2 children. What is the probability that the other child is his sister?
 A: 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.
A: 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.
A: It is unambiguously correct ( assuming odds boy/girl 50-50 & male always king).
Only possible family structures: b,b / b,g / g,b - all family structures are equally likely ; therefore probability King having a Sister 2/3 (b/g ; g/b).
On the below :
" 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?"
1/2 - 1/2 ;
But this is not a different interpretation of the same question ; It is a different question !! Because if you randomously select a King, the odds of the King coming from a family with 2 boys rather than a family with one boy, is two times as big. ( and because we know there are two times more families with one boy and one girl ; we get >> b,g/ g,b/ b,b/ b,b AND it is 50-50) 
; or said in a different manner ; now it is possible that the King has an older brother - if the younger brothers would be excluded from the random drawing - it would be 2/3-1/3 again. 
Few notes for fun :
* Are the odds Boy/Girl 50-50 ?
* King could be one of identical twins - meaning that there are ( small) odds in favour of b,b ; making the likelihood a little lower ( non identical twins assume normal distribution ; and identical girl twins are not in the possible poule anyway)
Cheers,
Sytse
A: Reduced Sample space is {bb, bg, gb}. Are equally likely if assumption is made that elder son will be king. But if anyone can be king bb, bg, gb are not equally likely but P(bb) = 1/2
Sample space should be changed if we assume there is no discrimination with boy or girl and on the basis of elder or younger. 
{KB, KG, BK, GK} that's reduced Sample space according to given condition. Without the given condition that ruler was king or queen.
P(Male) = 1/2 is assumed 
P(Elder is ruler) = 1/2 is assumed
Sample space is
{KB, BK, KG, BQ, GK, QB, QG, GQ}
K means male who is king Q is female who is Queen}
P(sibling is female) can be solved by considering.
Sample space = {KB, BK, KG, GK}
Event = {KG, GK}
P = 1/2
