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**Zennie** 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 $\displaystyle Q(B) = P(B|G)$; 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: $\displaystyle P(A) = P(A|B)P(B) + P(A|B^c)P(B^c)$

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.