boys and girls in a family

**kaka87** · November 12th 2009, 07:23 PM

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?

**Twig** · November 12th 2009, 09:38 PM

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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