Imagine we have $N$ molecules that are distributed randomly into $K$ boxes. What is the probability that a given box contains at least $m$ molecules? (For example, we could have $N=10K$, $m=20$ and we can think of $K$ as large.)
I'm not sure how to approach this problem... I'm thinking that we can model this as a multinomial distribution. We know that on average each box should contain $\frac{N}{K}$ molecules, but I don't know if it helps at all. From the example, I also think that we can consider a limiting behaviour since we know $K$ is large.
Assuming that each of the $N$ molecules is equally likely to end up in any box, and this is independent for each of them, the distribution of the number $X$ of molecules in each box is binomial, with parameters $(N,p)$ where $p=1/K$. Hence the probability that a given box contains at least $m$ molecules is $$ P(X\ge m)=\sum_{i=m}^N {N\choose i}p^i(1-p)^{N-i}. $$ Note that if $N,K\to\infty$ such that $N/K\to\lambda>0$, then the limit is a Poisson distribution with parameter $\lambda$.
For the Poisson approximation, you will get $$ P(X\ge m)\approx\sum_{i=m}^\infty \frac{\lambda^i e^{-\lambda}}{i!} =1-\sum_{i=0}^{m-1} \frac{\lambda^i e^{-\lambda}}{i!} $$ which is $0.00345...$ in your example.