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

. Find the expected number of boys in this family, and the expected number of girls.

Let

denote the number of boys in the family.

Let

denote the number of girls in the family.

We are seeking

and

.

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

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

Let

be any of the events in

.

Hence,

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

Thanks!