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Math Help - Conditional Probability using the Law of Total Probability

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    Conditional Probability using the Law of Total Probability

    From families with three children, a child is selected at random and found to be a girl. What is the probability that she has an older sister? Assume that in a three child family all sex distributions are equally probable.

    The book then gives this hint: Let G be the event that the randomly selected child is a girl, A be the even that she has an older sister, and O, M, and Y be the events that she is the oldest, the middle, and the youngest child respectively. For any subset B of the sample space let Q(B) = P(B|G); then apply the Law of Total Probability to Q.

    So I have the given information in the hint to start with and the possible combinations of three children. {bbb, bbg, bgb, gbb, bgg, gbg, ggb, ggg}.

    The Law of Total Probability states: P(A) = P(A|B)P(B) + P(A|B^c)P(B^c)

    I'm having a lot of trouble figuring out how to set this up and how to get started. I'm even questioning if I know what P(G) is for sure.
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    Quote Originally Posted by Zennie View Post
    From families with three children, a child is selected at random and found to be a girl. What is the probability that she has an older sister? Assume that in a three child family all sex distributions are equally probable.

    The book then gives this hint: Let G be the event that the randomly selected child is a girl, A be the even that she has an older sister, and O, M, and Y be the events that she is the oldest, the middle, and the youngest child respectively. For any subset B of the sample space let Q(B) = P(B|G); then apply the Law of Total Probability to Q.

    So I have the given information in the hint to start with and the possible combinations of three children. {bbb, bbg, bgb, gbb, bgg, gbg, ggb, ggg}.

    The Law of Total Probability states: P(A) = P(A|B)P(B) + P(A|B^c)P(B^c)

    I'm having a lot of trouble figuring out how to set this up and how to get started. I'm even questioning if I know what P(G) is for sure.
    The restricted sample space is:

    G1 G2 G3
    G1 G2 B3
    G1 B2 G3
    G1 B2 B3
    B1 B2 G3
    B1 G2 B3
    B1 G2 G3

    For the selected girl to have an older sister:

    G1 G2 G3: Either G1 or G2 is selected: 2/21
    G1 G2 B3: G1 is selected: 1/21
    G1 B2 G3: G1 is selected: 1/21
    G1 B2 B3: Selected girl has no older sister: 0
    B1 B2 G3: Selected girl has no older sister: 0
    B1 G2 B3: Selected girl has no older sister: 0
    B1 G2 G3: G2 is selectd: 1/21

    So I get 5/21.
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