[dchz]

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Interesting proposition. Lets say that you'd have to pitch in 3$ to play. Would this mean negative profit?

According to the EV=1 statement, it would.

However, you always win at least 2$.
So, you always win 2 or 4 or 8 or 16... and so on. So the EV has to be more than 2.

I'd say that the EV is a bit more than 2 with high positive variance. Just by thinking about cases how it could end up.

Think about it this way.
Would you play it if your bet would be 3$?

Also, lets assume you play it 100 times. Assuming perfect distribution you would end up with:
50 times you win 2$
25 times you win 4 and so on.
If we stop the flips at 5 tails and one head, which you'll get once you'd make ~600$. Divided by 100 games that makes 6$ per game.

Repeat that, but now 10 000 times. Assuming you never get to more than 9 tails and one head (around 1/10000) you'd make ~10 $ per game, thus you'd make about 100 grand.

I am not good at this kind of math (sums over ranges) but I think you could simulate the EV with monte carlo simulation for several amounts of games. Like once for 10, 100, 1000, 10000 and so on, and get some kind of result.

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didn't really read your post becuase your avatar gave me night mares

i think the EV is infinity but the utility of this is less and it's the paradox

i hope this enlightened someone cause it sure made my head spin