Probability - family of two children

[CAF123]

Homework Statement

A king comes from a family of two children. What is the prob that the other child is his sister?

The Attempt at a Solution

I have the correct answer below, I just want to check my argument is good. I think the question is badly worded because of the use of the word 'sister'. In order to get the correct answer, I believe you have to assume this means 'girl', otherwise you would have to consider whether or not the other child is the king's brother, sister, cousin, step-cousin etc..

Anyhow, what I said was P(other child girl|one child is a boy) = P(other child girl and one child boy)/P(one child is a boy).
Writing down the possible ways: BB, BG, GG gives the numerator as 1/3 and the denominator is 1/2 since we have either BG or BB. (1/3)(1/2) = 2/3.

Is this ok? It gives the correct answer but I am not sure if I can assume sister just means girl.

 

[bigfooted]

Assuming it doesn't matter if the sister is older or younger, and that the probability of boy/girl is 1/2, you either have:
first child = girl, second child = girl
first child = girl, second child = boy
first child = boy, second child = boy
first child = boy, second child = girl

Given that one child is a boy, so we are left with:
first child = girl, second child = boy
first child = boy, second child = boy
first child = boy, second child = girl
So the probability that the other child is a girl, given that one child is a boy is 2/3

 

[Ray Vickson]

Homework Statement

A king comes from a family of two children. What is the prob that the other child is his sister?

The Attempt at a Solution

I have the correct answer below, I just want to check my argument is good. I think the question is badly worded because of the use of the word 'sister'. In order to get the correct answer, I believe you have to assume this means 'girl', otherwise you would have to consider whether or not the other child is the king's brother, sister, cousin, step-cousin etc..

Anyhow, what I said was P(other child girl|one child is a boy) = P(other child girl and one child boy)/P(one child is a boy).
Writing down the possible ways: BB, BG, GG gives the numerator as 1/3 and the denominator is 1/2 since we have either BG or BB. (1/3)(1/2) = 2/3.

Is this ok? It gives the correct answer but I am not sure if I can assume sister just means girl.

This is a classic puzzle, and your answer is correct under certain circumstances: it is OK if you know essentially nothing about the King's sibling. However, if you are told the King is the oldest of two children, then the answer is incorrect; if you are told the King is the youngest of two children, the answer is also incorrect.

In actual royalty situations a King would typically be the oldest male child, so that would constitute some extra information and so might change the answer, but I have not thought it through.

The problem is analogous to being told that in two tosses of a coin you get at least one 'head', and you then want the conditional probability that the other toss is also a 'head'.

 

[CAF123]

@RGV Incidentally, the question about someone being older was the next question I tried.

 

[haruspex]

In actual royalty situations a King would typically be the oldest male child, so that would constitute some extra information and so might change the answer, but I have not thought it through.

No, that's still equivalent to 'at least one is male'. But it would change if the monarch is always the eldest sibling, regardless of gender (as in Sweden, Netherlands, Norway and Belgium):
1st born 2nd born
M M King with brother
M F King with sister
F M Queen with brother
F F Queen with sister

So to get 2/3, you need either that the monarch is randomly chosen from the siblings (never heard of such a practice) or that the male always takes precedence:
1st born 2nd born
M M King with brother
M F King with sister
F M King with sister
F F Queen with sister