We have $n$ locks with $n$ keys numbered $1$ to $n$. Locks and keys are packed together using the following process:
Pick up lock number $1$. Pick up a random key from the batch of $n$ keys. Pack them together. Then keep on picking up locks in the order $2, 3, \ldots, n$. Look for the key with the same number on the lock. If found, pack the matching set together. Otherwise, pick a random key from the remaining keys in the batch, and pack them together.
What is the probability that Probability that the lock numbered $n$ is packed with the key numbered $n$?
