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 probability that one particular ball is never drawn in $n$ tries is:
$$P(A_n) = P(B_n)= P(C_n) = {(\tfrac 2 3)}^{n}$$
The probability that two particular balls are never drawn in $n$ tries is:
$$P(A_n\cap B_n) = P(B_n\cap C_n)= P(A_n\cap C_n) = {(\tfrac 1 3)}^{n}$$
The probability that three particular balls are never drawn in $n$ tries is:
$$P(A_n\cap B_n\cap C_n) = {(\tfrac 0 3)}^{n}$$
So $P(X>n) = \frac{\cdot 2^n - 1}{3^{n-1}}$
Then $P(X=n) = P(X>n-1)-P(X>n)$
