Hashing Probabilities
For sufficiently large $M$, the size distribution of hash slots with good uniform hash functions follows a Poisson distribution.

Let $\lambda = \frac{N}{M}$ be the load factor. Then the expected proportion of buckets with exactly $k$ items in it is:

$$P(\hbox{# of items in the bucket is } k) = \frac{e^{-\lambda} \lambda^k}{k!}$$