Cards are dealt, one by one, from an ordinary deck, until the first diamond appears or 3 cards are dealt in total (whichever comes first). Find the distribution of the total number of clubs dealt.

What ive done so far

I am not exactly sure how to do this, but i started by breaking down into 4 scenarios.

1. First card is a diamond

2. Second card is a diamond

3. Third card is a diamond

4. No diamond

For the following, D = Diamond, C = Club, and X = Spade or Heart

1. D

Pr(C) = 0

2. CD, XD

Pr(C) = 1/3 (since x can be two different suites)

3. CCD, CXD, XCD, XXD

Pr(C) = 5/9

4. CCC, CCX, CXX, XXX, CXC, XCC, XXC, XCX

Pr(C) = 19/27

Would I now find the number of clubs possible in each scenario? And that would be the distribution?

for example, in scenario 3...

Pr(C = 1) = 4/9

Pr(C = 2) = 1/9

Let me know. Thanks for any help