Generalization of urn problem

While simple urn problems - given x
green and y black balls, whats the probability if we take 6 that 2
will be black etc. are easily solvable with hypergeometric and binomial distribution. Is there a distribution that solves it a general case for example:
Given 1000
blue, 1200 green, 2000 red balls we draw 20 balls.
Let x be the number of drawn blue balls currently, y-green, z-red(all currently drawn). If at any draw this condition is satisfied: y^3-z^2>x^2.5 . We add ONCE to the urn this configuration and continue the drawing
we add to the urn:
  1. The number of green balls drawn as green
  2. Twice the number of red balls as red
  3. Cubed the numbers of blue balls as blue.
What is the probability that at the end the red balls will be more than the green and the blue combined?
Is this solvable?


Ambassador to the humans
It might be directly solvable but it seems a bit too complex and calculations using simulation would be a lot easier to produce.