Find probabilities and probability mass function Suppose an urn contains three balls, once black, one white, and one green. Assume
that balls are repeatedly drawn, one at a time, with replacement, and let X denote
the number of draws until each colour appears at least once.
(a) Find the probabilities $P(X > n)$ for $n = 0, 1, 2,....$
(b) Find the probability mass function (pmf) of X  
For part (a) we can say $P(X>n) = P(A_n \cup B_n \cup C_n)$ where $A_n$ is the event that no black balls are drawn, $B_n$ is the event that no white balls are drawn and $C_n$ is the event that no green balls are drawn
So, $P(A_n \cup B_n \cup C_n) = P(A_n) + P(B_n) + P(C_n) - P(A_n \cap B_n) - P(A_n \cap C_n) - P(B_n \cap C_n) + P(A_n \cap B_n \cap C_n)$
How do I compute these probabilities ?

The case $n\leq 2$ is trivial. Assume $n > 2$: Following your notation the probabaility that you don't draw a certain ball until $n$ is $(2/3)^n$. $P(A_n \cap B_n \cap C_n)=0$ because you have to draw some balls. $A_n \cap B_n$ is the event that you draw $n$ green balls in a row. Did that help?
