Approximation with a Poisson distribution

It is clear that a direct numerical evaluation of $p_{0}$ is limited to the case of relatively small M and N. According to [28], it is possible to estimate $p_{m}$, the probability to have exactly m cells empty, with a Poisson distribution.

$$\lambda=M e^{-\frac{N}{M}}$$

$$p_{m}(N,M)= e^{-\lambda} \frac{\lambda^m}{m!}$$

In our case, we use $m=0$ to get all cells occupied:

$$p_{0}(N,M) = e^{-\lambda}$$