probability

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For a family with 2 children, the sample space indicating boy (B) or girl (G) is BB, BG, GB, and GG. If each of the outcomes is equally likely, find the probability that the family has 2 girls, given that the first child is a girl.

  • probability -

    This is a problem of conditional probability. The shortcut to the answer is consider only the outcomes where the first child is a girl, and calculate the probability among those where both are girls.
    I.E.
    Ω={GG,GB}
    favourable outcome: {GG}
    So assuming all outcomes are equally probable, then
    P(GG|first child is a girl)=1/2

    The proper way is to use the definition:
    P(GG|GX) ...both girls given first child is a girl
    =P(GG∩GX)/P(GX)
    =P(GG)/P(GX) .. since P(GG&GX)=P(GG) by absorption
    =(1/4)/(1/2)
    =1/2

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