[pzhon]

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So, the expected number of flips is [2]

So the excpected length of the game is 2 flips, and your EV is then 4.

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Non sequitur, and your conclusion is wrong. The correct answer has already been given along with justifications and links to other discussions.

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So my conclusion is wrong just because you don't agree?

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No, and that's not what I said. Your conclusion is clearly wrong regardless of how I feel about it.

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I stand by my calculations, the expected length of the game is two flips.

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Non sequitur means "that doesn't follow." You correctly calculated the average number of tosses, 2. That doesn't establish that the expected value is the second payout.

E(f(X)) is usually not equal to f(E(X)) when f is not linear and X is not constant. They are not equal here when you let X be the number of flips, and f(n) be the payout when the game lasts n flips.