[anifow]

If you are willing to spend a little time with it, this might help give you a more intuitive udnerstanding:

http://intromath.ca/aakkozzll/

The trick is to keep in mind that in this model, each ball has a 50/50 chance of either going left or right when it crosses each peg.

This is analogous to a game where you flip a coin 9 times in a row and win a dollar for every heads, lose a dollar for every tails. On average, you come out even, but the outcome of any particular game is like one of the balls falling down that Pachinko board. You might get really lucky and get 9 heads in a row, but usually there's a mix of heads and tails, and if you look at all the possible results, it would stack up like those balls in the pachinko board.