Last Puzzles (Hard Probability Question)

Imagine a situation in your mind. You are playing a coin game with your friend where you have pre-decided the sequence that you require. The coin is fair and you both keep tossing it till you get the desired sequence. The predefined sequence for you is H T H. The predefined sequence for your friend is H T T. You keep tossing the coin and writing it down till you get the required sequence. You play several games and at the end, the one with the average score less will win. Game 1 result: You: H T T H T H Your score: 6 Your friend: H T H H H T H H T T Your friends' score: 10 Game 2 result: You: T T H T T H H T H Your score: 9 Your friend: T T H H T H T T Your friends' score: 8 Game 3 result: You: T T H H T H Your score: 6 Your friend: H H T H T T Your friends' score: 6 Now, after playing 3 games, your score is 7 (average) and your friend's score is 8 (average). Assume that you keep playing like this for several many games, what of the following outcomes are you expecting at the end? a) Your average score is lower and you win. b) Your friend's average score is lower and he wins. c) Both have equal scores and the match ties.

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