You should not pay more than $0.50 to play this game. Here is why.
Each flip brings the odds of your slip draw closer to 50:50 odds. Yes you may hit heads on your first flip at which point, you take the casino for their dollar. This will only happen 50% of the time. The other 50% of the time you will get a tails which forces you to keep on flipping to try for a better chance than 0% on your paper slip drawing.
Say at some point you get 4 heads and 3 tails. That is a ~ 57% chance of winning that dollar (i.e. 4 heads / 7 slips). The longer you go the closer you get to 50:50 odds. Paying any more than $0.50 may win you money in the early stages, but only 50% of the time. The other 50% of the time you keep flipping to play catchup with the tails you have accumulated bringing you ever closer to the 50:50 odds you get from just flipping the coin in the first place :D
Reponse To Comments:
Someone asked if you can get above 50% by continuing to flip. You will most definitely get more heads, and for a short time may even have greater than 50% odds, but your best odds happen with early flips. The trick to this puzzle is you have to realize that the times when you are above 50% win rate only happens 50% of the time you play this game with the casino.
Take for example that I flip heads on my first flip. I have a 50% chance of doing that and it will give me a 100% chance on my win rate during the dollar draw. My real odds end up being 50% for the flip time 100% for the draw or 0.5 * 1 = 0.5 or 50% because the other 50% of the time I will hit a tails and have a 0% chance of winning the draw and have to keep flipping or forfeit my bet.
You can't do better than 100% chance of winning, so the other 50% of the time you either lose your entire bet or you keep flipping to get an advantage of 50% or greater. The kicker is as you keep flipping, 50% of the time you will dig the hole deeper (i.e. get more tails). The more you flip, the closer the odds converge to 50:50 because you can't outpace the tails you have accumulated and even when you do get a streak of heads going, you are equally likely and definitely will hit an equal streak of tails whether it be this round of betting or some future round since the coin flip is 50:50.
So even though I may have more heads (at some point in time), the ratio of heads to tails will always converge on 50:50.
3 heads / 4 flips = 75%
6 heads / 10 flips = 60%
30 heads / 55 flips = 54.5%
1000 heads / 1900 flips = 52.6%
1000 heads is a lot of heads, but 1000 when compared to 900 tails is a smaller advantage than when you compare 6 heads out of 10 flips.
And I know someone out there is thinking, well I would just stop at the 75% and play (3/4), but remember that is just a rate at which you CAN win a dollar. The other 25% of the draw will lose your bet. Even worse, in a future round of betting you ABSOLUTELY WILL come across a game where you have 3 tails and 1 head giving you only a 25% to win the draw and 75% chance to lose. In this scenario, you can either play catch-up with the the tails, which can take you deeper into the tail territory or in the best case even you out to the 50% win rate or you can choose to submit to a win rate lower than 50% in which case you negate any winnings you had in previous rounds of betting.
In the long run you will never come out above a 50% win rate and should not pay more than $0.50 to play.