Suppose we throw $n$ indistinguishable balls in $n$ bins at random. The throws are independent. What is the expected number of empty bins? What is the expected number of bins with one ball.
Using indicator random variables, expectations, some sloppy math and some questionable logic, I arrive at the conclusion that both are approximately $n/e$. I'm not able to find simple formulas. It also sounds very counter-intuitive to me.
Worse, I wrote a small Ruby program to simulate it and the results appear to be (approximately) correct.
Can anybody show me a good solution?