No, you have to state N before you start flipping coins.
Yeah, which means if I'm trying to maximize my payout, I'll set N arbitrarily large and abort the game at sufficient evidence that the coin isn't predictable enough for the game to have positive expected value. If the coin is predictable enough, then I'll pump my friend for every last cent he has.
However, note that the problem as stated asks for the minimum value of N so that the game has positive expected value. (I'm not too sure why we're interested in this except as an exercise).
edit: just clarifying for others. Not that I think you misunderstood.
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