Assume that all heads:tails ratios are equally likely for the coin.
Presumably there is still a fixed (frequentist) probability of a flip being heads, otherwise your coin either has memory or is a subject to an unknown external interference.
You have to accurately predict at least 75% of flips to come out ahead, there is no way to do that with a fair(ish) coin, so, even if you have a perfect model of the coin you will likely lose if the coin is moderately unfair (25%-75% heads), regardless of N.
Maybe I misunderstand your setup.
You can stop playing early if the coin appears too unbiased, to limit losses, but keep playing for the full N turns if the coin seems predictable enough.
I came up with this puzzle after reading Vaniver's excellent post on the Value of Information. I enjoyed working it out over Thanksgiving and thought I'd share it with the rest of you.
Your friend holds up a curiously warped coin. "Let's play a game," he says. "I've tampered with this quarter. It could come up all heads, all tails, or any value in between. I want you to predict a coin flip; if you get it right, I'll pay you $1, and if you're wrong, you pay me $3."
"Absolutely, on one condition," you reply. "We repeat this bet until I decide to stop or we finish N games."
What is the minimum value of N that lets you come out ahead on average?
Each game, you may choose heads or tails, or to end the sequence of bets with that coin. Assume that all heads:tails ratios are equally likely for the coin.
edit: since a couple people have gotten it, I'll link my solution: http://pastebin.com/XsEizNFL