A king has n children, at least one of them is a daughter. What’s the probability that all of them are daughters?

A king has n children, at least one of them is a daughter. What’s the probability that all of them are daughters?

So far I've considered the case of 3 children, which gives a probability of 1/7. But I'm confused about how to generalise this?

The sample space has $2^n$ equi-probable points, assuming P(boy) = P(girl) = $\frac12$

Knowledge that there is at least one girl implies that one particular outcome is ruled out.

Continue....

Assuming that the probability of a child being a girl is 1/2, which is not quite true biologically but close ...

Every child is either a boy or a girl. (Old fashioned idea, but I'm sticking with it.) So with n children there are 2^n possibilities.

If "success" is defined as all girls, then that is exactly 1 of the possible combinations. So, ignoring the "at least one girl" requirement for the moment, the probability of all girls is 1 / 2^n. e.g. with 1 child, 1/2; with 2 children, 1/4; 3 children, 1/8; 4 children, 1/16; etc.

But we don't want to consider all possibilities, but only those where at least one child is a girl. So the denominator should not be 2^n, but only those where at least one of the children is a girl. This is an easy special case, though: just as there's only one way for all the children to be girls, there's only one way for none of the children to be girls: they must all be boys. So of the 2^n possible combinations, the number that include at least one girl is 2^n-1.

So the probability is 1 / (2^n-1). e.g. 2 children, 1/3; 3 children, 1/7; 4 children, 1/15; etc. Note that if there's only one child, this formula gives 1/1=100%, which makes sense: If you have only one child, and at least one of them is a girl, then that one child must be a girl and they are "all" girls.

** Update **

Okay, apparently an unclear point here.

If there are two children, that gives 2^2=4 possibilities: BB, BG, GB, and GG. Note that there is only one way each to have all boys or all girls, but 2 ways to have one boy and one girl. So, considering all cases, not just the ones where at least one child is a girl, the probability of all boys is 1/4, all girls is 1/4, and one of each is 2/4 = 1/2.

Similarly with 3 children the possibilities are BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG. The probability of all boys is 1/8, all girls is 1/8, 1 boy and 2 girls is 3/8, and 2 boys and 1 girl is 3/8. That is, there's more than one way to have two boys and a girl -- the girl could be first, second, or third -- but there's only one way to have all boys or all girls.