A hotel has $n$ rooms numbered $1,2, \ldots, n$. For each room there is one spare key labeled with the room number. The hotel manager keeps all the spare keys in a box. Her mischievous son got hold of the box and permuted the labels uniformly at random. What is the expected number of keys which still open the room whose label they carry? [Hint: Use linearity of expectation]
- $1$
- $\frac{n-1}{n}$
- $\frac{n}{n-1}$
- $\frac{n}{2}$
- None of the above