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**Twig** Hi!

A family has children until it has a boy or until it has three children, whichever comes first. Assume the child is a boy with probability $\displaystyle \frac{1}{2}$. Find the expected number of boys in this family, and the expected number of girls.

Let $\displaystyle X$ denote the number of boys in the family.

Let $\displaystyle Y$ denote the number of girls in the family.

We are seeking $\displaystyle E(X)$ and $\displaystyle E(Y)$.

There are the following possible "child outcomes": G = girl B = boy

$\displaystyle \Omega = \left(GGG,GGB,GB,B\right) $

It is not possible to have for example BBG, since the family stops having children when they have gotten a boy.

Let $\displaystyle F$ be any of the events in $\displaystyle \Omega$.

Hence,

$\displaystyle E(X)=\displaystyle \sum_{\Omega} E(X|F)P(F) = \frac{3}{4} $

$\displaystyle E(Y)=\displaystyle \sum_{\Omega} E(Y|F)P(F) = \frac{3}{2} $

Did I go wrong somewhere, if so, what is the error?

Thanks!