Let's see. I'll assume there are enough balls in each urn that it can be picked from indefinitely. (that is to say each urn would have y balls)

If y = 1, then the chance is 0% (trivial, since you can't choose a different urn if you only have one)

If y = 2, then the possibilities are (let numbers be the colors)

1 1

1 2

2 1

2 2

so the probability is 1/2

If y = 3, then there are 27 possibilities. I counted 18 that weren't all a single color are possible, so the probability would be 2/3.

At this point, I smell an induction proof, and I'm sure there's an easier way with permutations and combinations.