# Is there a fast, reasonably accurate estimator for multinomial PDF?

I am working on a balls in boxes kind of problem, where the probability of a ball ending up in a certain box varies by box, that is, each box has some probability P of getting any ball, all together summing to 1 obviously.

I'm trying to answer "what is the probability that after X balls are assigned to K boxes that any box has more than Z balls?" kind of questions.

I my cases, the number of boxes can be large (>500) with number of balls ~10-20% of the box count. Now, when the probability for all the boxes is uniform, I can short-cut the problem and get results quickly by getting all possible combinations of integer partitions with a maximum of at least Z, calculating the probability of such a combination, and multiplying that by the number of permutations of that combination, finally summing. But when they differ, I have to enumerate each of the possible permutations, calculate the probabilities, and sum. This gets huge quite quickly.

Question: is there any kind of reasonably accurate estimator for this kind of probability question that does not require total enumeration?

Notation: $x$=number of balls, $u$=number of boxes (or urns as statisticians prefer), $N_i$ is the number of balls in box $i$ and $p_i$ is the probability that a ball lands in box $i.$
We know $(N_1,...,N_u)$ is multinomial with binomial maginals for each $N_i$. Of course, $N_i$ and $N_j$ are dependent but the more boxes we have, the less the dependence. The dependence also is a function of the $p_i.$
I worked out the correlation: $$\rho=-\sqrt{\frac{p_ip_j}{(1-p_i)(1-p_j)}}$$ If the probabilities are equal, $\rho=-\frac {1}{u-1} .$ Given such a small dependence, let's treat the $(N_i)$ as independent.
Then, for $z=1,2,...,x$ let $$F(z)=P(\max_{1\le i\le u} N_i\le z)$$ $$\approx \prod_{1\le i \le u}P(N_i\le z)$$ $$=\prod_{1\le i \le u} \sum_{m=0}^{z}{x \choose m} p_i^m (1-p_i)^{x-m}$$ Then $1-F(z)$ is the probability that at least one of the boxes contains more than $z$ balls.