Continuous Random Variable question, Probability and Statistics a little help please
A couple decide they really want a daughter. So, they decide to start having children and continue until they have their first daughter. Assuming having either a boy or girl is equally likely, answer the following:
(a) In the end, will the couple be more likely to have more boys or more girls? Explain why.
(b) Give a formula for the probability that they end up with exactly k boys. 
So for (a), I feel like the intuitive answer would be that the couple is more likely to have a boy. This is because once you have a girl, your done. So the first child is 1/2 likely to be a boy or girl. But, then you have to think about the cases in which multiple boys are born for a girl. {G, BG, BBG, BBBG, etc..} So if I add up all these probabilites, does the chance of having a boy outweight that of having a girl?  
For b, I feel like the geometric distribution is the distribution that I need. p(k) = ((1-p)^(k-1))p
This finds the kth success. So if we consider a girl k, then we can find the number of boys? I'm not exactly sure how to think about this, especially without knowing p

(a) Expectation on the girl is of course $1$. And for the boys we have something like $P\{B=k\}=\frac{1}{2^{k+1}}$, where $k\in\{0,1,...\}$, $B$- distribution that counts boys. $$\mathbb{E}[B]=\sum_{k=0}^\infty \frac{k}{2^{k+1}}=1.$$
It's exactly the $1/2$ that they have alone girl, and $1/2$ that girl+any positive natural number of boys. 
And the (b) answer has already been given.
