I am given $n$ balls. For $n$ times, I pick one of them with uniform probability and put it back after picking it.


Let $U$ be the number of balls I have never picked, so $U\in \{0,\ldots,n-1\}$. We are interested in the expected value $E\_n:= E(U)$.


What is the value of $\lim\_{n\to\infty} \frac{E\_n}{n}$, if it exists?


(This is a follow-up question to [this question](https://mathoverflow.net/questions/257047/choosing-n-times-from-n-objects).)



Expected number of balls left out when choosing $n$ times from $n$ balls
When tossing $n$ balls uniformly and independently into $n$ bins, the distribution of the number of balls in any specific bin is binomial with parameters $n$ and $p=\frac{1}{n}$. Thus, the probability that a bin is left empty is $(1-\frac{1}{n})^n$, which tends to $\frac{1}{e}$. By linearity, the expected number of empty bins is $n (1-\frac{1}{n})^n \approx \frac{n}{e}$.


