What is the probability that the first bin is empty?

Question

Suppose that n balls are tossed, one at a time, into n bins such that each ball is equally likely to fall into any bin and that the tosses are independent. What is the probability that the first bin is empty?

Answer 1

Toss one ball and the chance that it will miss the first bin is $p=\frac{n-1}{n}$. But you have to miss the first bin $n$ times in a row so the probability is:

$$P=p^n=\left(1-\frac 1n\right)^n$$

It's interesting to notice that for big values of $n$ the probability is approximately $\frac1e$.