I have $N$ sets of differently colored balls, each set a distinct color and each set consisting of 3 balls. There are 9 bins.

Balls are tossed and will land into bins uniformly randomly, except if that bin already has a ball of the same color, whereby the tossed ball will land in another bin devoid of that ball's color with uniform probability.

In addition, no balls of the same color can occupy bins 6 and 7 or bins 8 and 9 simultaneously, that is, for a given color ball, it cannot have balls in bin 6 and 7 at the same time, the same for bins 8 and 9. A same colored ball in say bins 6 and 9 is allowed, as is true for bins 7 and 8 - think of it as the first five bins are in a row, the last four bins are stacked two high, giving 7 bin positions left to right, and only one ball of a given color is allowed in a given position.

I'm trying to get to the probability that once all the balls are tossed, at least one bin of some subset of the 9 (in my case of interest, bins 1 through 5 and bins 8 and 9, the "bottom" bins of the two stacked pairs) is empty.

I could do this using the PDF of a sum of $N$ multivariate hypergeometric distributions, but such a thing does not seem to exist based on my Google-fu.

Does such a thing (PDF for the sum of $N$ multivariate hypergeometric distributions) exist, or is there some more direct way to do this that eludes me?

Edit: After proofreading, I realized the PDF idea is a no-go: it would not account for the stacked bins exception... ideas?


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