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)

means.

Look at

http://www.mathhelpforum.com/math-he...-question.html

because most likely, you are sampling without replacement and it just another hypergeometric.