For the purposes of this problem, a linear Pachinko machine is a sequence of one or more of the following: holes ("."), floor tiles ("_"), walls ("|"), and mountains ("/\"). A wall or mountain will never be adjacent to another wall or mountain. To play the game, a ball is dropped at random over some character within a machine. A ball dropped into a hole falls through. A ball dropped onto a floor tile stops immediately. A ball dropped onto the left side of a mountain rolls to the left across any number of consecutive floor tiles until it falls into a hole, falls off the left end of the machine, or stops by hitting a wall or mountain. A ball dropped onto the right side of a mountain behaves similarly. A ball dropped onto a wall behaves as if it were dropped onto the left or right side of a mountain, with a 50% chance for each. If a ball is dropped at random over the machine, with all starting positions being equally likely, what is the probability that the ball will fall either through a hole or off an end?
For example, consider the following machine, where the numbers just indicate character positions and are not part of the machine itself:
123456789
/\.|__/\.
The probabilities that a ball will fall through a hole or off the end of the machine are as follows, by position: 1=100%, 2=100%, 3=100%, 4=50%, 5=0%, 6=0%, 7=0%, 8=100%, 9=100%. The combined probability for the whole machine is just the average, which is approximately 61.111%.
/\.|__/\.
_._/\_|.__/\./\_
...
___
./\.
_/\_
_|.|_|.|_|.|_
____|_____
#
61
53
100
0
100
50
53
10