We're going to call this three chances instead of coin flips, because at two-to-one odds, it's obviously not a coin.
The odds of three "heads" is 2/3 x 2/3 x 2/3 = 8/27.
The odds of three "tails" is 1/3 x 1/3 x 1/3 = 1/27.
This leaves 18/27, which split 2-to-1, 12/27 for two "heads" and 6/27 for two "tails".
Translated to money, in 27 trials you can expect: (8 x $300 = $2400) for three heads, (12 x $200 = $2400) for two heads, (6 x $100 = $600) for two tails.
This adds up to $5,400, an average of $200 per play. If you have to pay more than that, you will lose money in the long run.
If you change the game by making it a zero result if the first two chances come up "tails," that only affects one third of the two-tails chances, the ones with tails-tails-heads. That's two of six, which change from $100 winnings to nothing.
Subtract $200 from the $5,400 total, and divide by 27 as before. Is the game still worth $200 to play?
It is important to state that probability dictates there is no such thing...
(The entire section contains 7 answers and 579 words.)