An event has probabilitypof success andq(=1-p) of failure. Independent trials are carried out until at least one success and one failure has occurred. Find the probability thatrtrials are necessary and show that this probability equals when .

A couple decide that they will continue to have children until they have both a boy and girl or they have four children. Assuming boys and girls are equally likely to be born, what will be the expected size of the completed family.

Sum of the general term =

when

So I have gotten these parts right, but about the last part, how do I do it?

Because I suppose I use to find the probability of having a boy and girl after r trials. So I find the value when r is 2, 3, and 4? and then each of those multiply by the number of children?

Thanks