Yup I know it's an old thread... ^^
I'd say that your approach is not correct. Because you're considering that you get the "double red" sucessively. While you don't take into account the case :
1st draw : double red.
2nd draw : red & other
3rd draw : red & red
and also, you haven't considered that X=2 means that there are exactly two "double red".
and by the way, X has to stop at
I don't know how to explain it more clearly, but what I can say is that I think (almost sure - with a probability 99% ) your reasoning is not correct.
Two possible approaches I can think of :
- cumulative density function
- conditional expectation
But I don't know if one of them leads somewhere... as it is not that easy to represent the problem But that's a good way of bumping, eh ?