Circled Erasure

The numbers $1,2,3,\dots,n$ are written on a whiteboard and then erased via the following process:

Before any numbers are erased, a pair of numbers is chosen uniformly at random and circled. For the next $n-1$ minutes, a pair of numbers still on the board is chosen uniformly at random each minute, and the smaller one is erased. In minute $n$ , the last number is erased.

What is the probability that the smaller circled number is erased before the larger? Report the answer with $n=1000$ .

Answer format: round to 3 decimal places.