Rule of thumb when drawing N samples from a discrete distribution with N possible values with replacement I'm looking for an explanation and possibly the name of a rule of thumb:
When drawing N samples with replacement from a discrete uniform distribution of N values, it is very likely that:


*

*1/3 of the values are not drawn

*1/3 of the values are drawn exactly once

*1/3 of the values are drawn more than once


The way I heard that was with throwing 64 pieces of rice onto a chess board and then observing roughly what I described above. I'm seeing this very often, but I want to read up on the theory and calculations behind it.
Why is this rule of thumb valid and does this rule of thumb have a name? And if yes, which one?
 A: The rule is called law of small numbers and was described by Ladislaus Bortkiewicz (German: Gesetz der kleinen Zahlen from Ladislaus von Bortkewitsch).
Credit goes to whuber for pointing me to the Poisson distribution. I'll leave it to him to provide a better answer.
In short: According to the Poisson distribution, the amount of object that get "hit" exactly $k$ times for $n \rightarrow \infty$ is
$$P(X=k) = {1 \over k!} e^{-1}$$
This means the amount of values that are never drawn ($k=0$) is ${1 \over e} \approx 36,7879 \%$ and the same for ($k=1$), which is approximately one third.
A: Yes, there is a rule, but it's much more precise than a rule of thumb. This type of problem is modeled by a multinomial distribution, and it actually models something even more general than what you have described: suppose that instead of drawing $N$ samples with replacement from a population of $N$ distinct values, you draw $n$ samples from a population of $k$ distinct values.  Thus, it even models the situation where the number of samples and the number of values aren't necessarily equal.
