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Probability - family of two children

  1. Nov 29, 2012 #1

    CAF123

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    1. The problem statement, all variables and given/known data
    A king comes from a family of two children. What is the prob that the other child is his sister?

    3. 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.
     
  2. jcsd
  3. Nov 29, 2012 #2
    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
     
  4. Nov 29, 2012 #3

    Ray Vickson

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    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'.
     
  5. Nov 29, 2012 #4

    CAF123

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    @RGV Incidentally, the question about someone being older was the next question I tried.
     
  6. Nov 29, 2012 #5

    haruspex

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    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
     
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