But you can't control when you win. Okay, say you win the first 2 rounds, you got $142.87 now. You lose the next round, you got $114.29 left. You lose again, you got $85.71 left. You win the next round and thus the game, but your purse of $85.71 means that even if you bet it all, you still can get to the goal of $200. This is even more apparent if your coin out come is head, tail, head, tail, head. You'll be +14.29, -28.58, +14.29, -28.58 + 14.29 < 100.
I agree with the guy that said wait for the first two heads to come out, all in on the third. 50/50 odds (which is the best you can do with a coin toss) and you're guaranteed the extra $100 if you win the game.
I just posted a quick correction since I misstated the loss condition: you lose on the 3rd tail. You can't guarantee to win the game (in the terms of getting three heads). But you can guarantee to get the payoff if you win. The problem isn't about how to get win the game (you can't). It's about how to structure your bet to get your desired payoff.
I think you are missing the point (or I'm explaining it badly). The point is to figure out how to bet so that if you win the game, you are guaranteed to double the game. So, you never want to be in a situation that you've won the game, but you haven't made $100 on top of your initial stake. There is a solution to this problem.
Before you clarified that was a brain teaser of monstrous proportion as I was trying to figure WTF the rules are.
He stated to devise a strategy. For example, if you got 11 versus 011 or 0011. Betting $50 each time would not make sense.