The definition of independent probabilities is that irregardless of the conditions in that probability, the probability remains the same.
In symbols. Events A & B are independent if P(A|B) = P(A)
and P(B|A) = P(B).
Successive flips of a (fair) coin are independent. That is, it does not matter how many heads there has been in a row, the next flip P(H) = 0.5 and P(T)=0.5. Next time you go to the casino you can no confidently make fun of the people who bet on the roulette table "Oh, there have been 4 blacks in a row, so red MUST be coming up."
Sampling of the balls in your example can be independent if you replace the drawn out ball after each pick. With replacement, the probabilities remain 4/7, 2/7, and 1/7 for R,B,G respectively. The conditional probability in my 1st response has to be used since you gave additional information that the ball drawn out is not red. So, you have to look at the subpopulations with the given information that the drawn ball is not red.
If you sample the balls without replacement, then once again you have to start using conditional probablilties. For example, we can calculate all the probabilites if the 1st ball is red. If the 1st is R, then the remaining population is 3R, 2B, 1G. So, the probabilites in order are 3/6=1/2, 2/6=1/3 and 1/6.
so Probability of drawing a red ball given that the 1st drawn ball is red=
P(R|R) = 1/2
P(B|R) = 1/3
P(G|R) = 1/6
and so on for all the cases.
Finally, a lot of these things are explained in introductory probability books. There are a lot of good ones out there, with lots more examples than just this. I'd encourage you to go check them out if you want to understand this al lot more fully.