Ladder lottery: how many rungs until it forgets?

A ladder lottery — amidakuji (阿弥陀籤) in Japan, sadaritagi (사다리타기) in Korea, guijiaotu (鬼腳圖) in China — is seven vertical lines with horizontal rungs thrown across them. Start at the top of one line, walk down, and take every rung you meet. Where you finish is fixed once the board is drawn: there is nothing random about the walk. The randomness is in the board.

So here is the question this page is about. If the board is drawn by someone else, how much does your choice of starting line still matter? Turn the knob and find out.

You start at1
Boards drawn2000
Gap from even0.00
One board drawn at random with that many rungs. Click a number at the top to start somewhere else. The purple line is your path.
Where you finish, over two thousand freshly drawn boards with the same number of rungs. The dashed line is one seventh — a dead-even finish.