If $p_n$ denote the probability that when $n$ balls are randomly put in $n$ bins then there is at least one bin with exactly one ball. Is there a simple (involving only little computation) reason for why $p_{n+1}>p_{n}$ if $n>3$ ? This simple looking problem turns out to be not too simple, perhaps because ${p_{n}}/{p_{n-1}}$ turns out to be approximately $e(1-{1}/{n})^{n-1}$ and hence whatever the difference is, it is extremely small.
Thanks