The probability of each ball being picked is , which is a start.
The way I would solve this would be to find the probability of two red balls, the probability of two green balls, and the probability of two white balls. If you add these, you will find the probability of receiving two balls that are the same colour.
I'm going to assume (as you haven't specified) that the balls are not replaced, and that this is therefore conditional probability.
P(2 red) = . This is because there are red balls out of the total during the first selection, and then there are red balls left out of the remaining balls during the second selection.
P(2 green) is the same as p(2 red) because there are the same amount of red balls as green balls, so p(2 green)
P(2 white) =
Do you understand so far?
So what is the total probability of receiving two balls that are the same colour?
And, therefore, how can you work out from that the probability of receiving two balls which are not the same colour? Show your working if you get stuck.