A man picks up a huge circular dart board and shoots paintballs at it. $n$ paintballs manage to hit it. They are random and point-sized, for the purpose of this question. A bug is somewhere on the edge of the board.
It likes to eat paint, so it goes to the nearest smear of paint and eats it. It then goes to the nearest smear from there and eats that also. It repeats this process until they entire board is clean. Incidentally, the bug also leaves a trail behind, wherever it goes.
Question: How many times is this trail most likely to intersect itself? (Report as a function of $n$)