It all depends on what you do with the first ball.
If you replace it, then this is a multinomial distribution.
If you do not replace it, then it's a hypergeometric distribution.
I don't know what
P(x,y) = P(x).P(x|x)+ P(x).P(y|x)+ P(y).P(y|y)+ P(y).P(x|y)
because most likely, you are sampling without replacement and it just another hypergeometric.