[pococurante]

Whether I change the game or not, the point is still the same. You can't get the EV by calculating the average # of flips.

Let's play the game 1024 times. Results:

512 times the game will last 1 flip = 512x$2 = $1024
256 times the game will last 2 flips = 256x$4 = $1024
128 times the game will last 3 flips = 128x$8 = $1024
64 times the game will last 4 flips = $1024
32 times the game will last 5 flips = $1024
16 times the game will last 6 flips = $1024
8 times the game will last 7 flips = $1024
4 times the game will last 8 flips = $1024
2 times the game will last 9 flips = $1024
1 time the game will last 10 flips = $1024
(final game with infinite value is being ignored here)

Results: In 1,024 games, a total of $10,240 was paid out.

Average number of flips per game: 2
Average payout per game: $10