Q:

Application of conditional probability

I was faced with the following problem:
"Calculate the probability of king's oldest sibling being female, when there are four siblings. Solve this conditional probability in two different cases of succession: a) Crown passes always to the oldest child, b) Crown passes to the oldest son, or if all children are girls, to the oldest child. (Genders of the children are independent, and the probability of each children being a girl is 1/2.)"
Now, in the first case the king is obviously oldest sibling, since the crown always passes to the oldest one. But what about in case b? I'm not very good at conditional probability, so little help is appreciated.

A:

The previous king and queen had 4 children.  That would suggest that there were $2^4=16$ possible combinations of genders.  But since we know that there is at least one boy, all girls can be removed as a possibility.  There are $15$ combinations of genders.
case a)
The king is the oldest child and he is male.
There are 8 remaining possible combinations of genders for the remaining 3 siblings, and there are 4 combinations for the oldest sibling to be female, and 4 for the oldest sibling to be male.
The kings oldest sibling is female $P = \frac {4}{8}$
case b)
If the kings oldest sibling is female, either the oldest child is male and the second child is female, or the oldest child is female, and one one the next 3 is male. 
If the oldest child is female there are 7 possible combinations of genders for the next 3.
The kings oldest sibling is female is $P = \frac {7+4}{15}$

