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Math Help - boys and girls in a family

  1. #1
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    boys and girls in a family

    Suppose a family has three children (consider all combinations -- there are eight of them), and a child picked at random from all children of such families. If he is a boy, what is the probability that he has two sisters?

    In this case, I did the following:

    P(Boy random | Two sisters in a family) = P(Two sisters in a fam. | Boy random)* P (Boy random) / P (Two sisters in a family). (Bayes Theorem.)

    The answer is supposed to be 1/4. However, I got 1/3. Do you guys see why that is?
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  2. #2
    Senior Member Twig's Avatar
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    Hi!

    Given that a boy was chosen, what is the probaility he has two sisters?

    P(\mbox{ Two sisters }|\mbox{ Boy chosen })=\frac{P(\mbox{ Boy chosen }|\mbox{ Two sisters } )P(\mbox{ Two sisters } )}{P(\mbox{ Boy chosen })} = \frac{\frac{1}{4}\left(0+\frac{1}{3}+\frac{1}{3}+\  frac{1}{3}\right)\frac{1}{2}}{\frac{1}{8}\left(1+\  frac{2}{3}+\frac{2}{3}+\frac{2}{3}+0+\frac{1}{3}+\  frac{1}{3}+\frac{1}{3}\right)} =\frac{1}{4}
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