Had to google pachinko, and I guess roulette is sorta like that.
The actual game I’m playing is a city building flash game, and you need to insert 9 “runes” into the slots of a ring on an ancient wonder to complete it.
That’s approximately 0.0937% chance to win after 9 plays.
Directly computing the average is a nightmare. However, if you disregard the “rule of 1 win after 10 fails”, you can linearize the expected value to simply get:
1 + 9/8 + 9/7 +9/6 +… +9/2 + 9/1
~=
25.5 times