A sequence of events

[kurosh]

There is an event that happens 1/10 times. It costs $1 for each attempt. If the attempt succeeds, you get $9 back, for a profit of $8. If you play up to 10 times, but stop when you win, is it profitable?

I know this shouldn't be profitable, but let me explain some logic and tell me why it's faulty?

In a sequence of 10 events, NOT STOPPING, the event will be successful once. In this sequence, it will evenly be distributed throughout each try, averaging at try #5.5.

Visual Display: X = hit
10 trials of 10 sequences
X _ _ _ _ _ _ _ _ _
_ X _ _ _ _ _ _ _ _
_ _ X _ _ _ _ _ _ _
_ _ _ X _ _ _ _ _ _
_ _ _ _ X _ _ _ _ _
_ _ _ _ _ X _ _ _ _
_ _ _ _ _ _ X _ _ _
_ _ _ _ _ _ _ X _ _
_ _ _ _ _ _ _ _ X _
_ _ _ _ _ _ _ _ _ X

This is average, yes? So if you go through one sequence, stopping when you hit, it should be profitable?