Is there an intuitive explanation for the probability mass function of Y that you discovered?

Suppose I have an urn with 9 balls: 4 green, 3 yellow and 2 white ones. I draw a ball from
the urn repeatedly with replacement, until I see the first green or yellow ball, and then I stop. Let
N be the number draws I needed. Let Y equal 1 if the last draw is green and 2 if the last draw is
yellow. Find the joint and marginal probability mass functions of N and Y and determine whether
N and Y are independent. Is there an intuitive explanation for the probability mass function of Y

that you discovered?

I'm completely loss and can't even start. From what I understand, $N \sim Geom(\frac{7}{9})$, but I can't get what is the distribution of Y. What is the pmf of Y? And how to find joint pmf? Thank you.

Answer

Guide:

Note that\begin{align}
Pr(N=n, Y=1) &=\left( \frac29\right)^{n-1}\frac{4}{9}
\end{align}

Finding $Pr(N=n, Y=2)$ should be similar and you should be able to compute the marginal distribution.