You are playing a probability game with your friend using a fair coin. Both of you decide a particular sequence that you have to achieve.
Let us suppose you chose the sequence to be: H T H
Your friend chose the sequence to be: H T T
Now you keep tossing coin until you get the sequence and the same is done by your friend. You keep doing that till you achieve your predefined sequence and keep writing the result on paper. At the end of the game the player whose average number of tosses will be lowest, he will win.
The results of game 1 toss:
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
The results of game 2 toss:
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
The results of game 3 toss:
You: T T H H T H
Your score: 6
Your friend: H H T H T T
Your friends' score: 6
Now after 3 games, your average score is 7 and your friend's average score is 8. Now assume that you keep playing the game and play many a times. What will be the possible outcome out of the following?
a) You win
b) Your friend win
Analyzing thing, the friend has better chances of winning. Let us assume and analyze what can happen at any point of time supposing that we have not already completed the sequence. If the last flip was tails, start at 1 and if it is heads, start at 2.
1) If any one between us flips to get tails, nothing will change. We will still be at step 1 and we both will be needing 3 additional flips to complete. But if either of us gets heads, we continue.
2) If either of us gets heads, we stay on this step. But if we get tails, we continue. At this point, I am 3A and my friend is 3B.
3A) If I get tails now, I will have to start all over again from step 1 till I again get heads and I will have to flip the coin at least 3 times to complete my sequence.
3B) If the friend gets heads, he goes back to the step 2 and try again for tails. He will have to flip the coin at least 2 times to complete his sequence.
Thus, he obviously has better chances of winning.