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.
Solution:
After analyzing the possibilities, it is obvious that my friend has better chances of winning and his average score will be lower at the end.
Let us assume that we have not completed the sequence yet and understand the possibilities. If the last toss of coin was tails, start with step 1 otherwise start at step 2.
Step 1: If either of us toss the coin to get tails, nothing will change. We both will still be at step one and we both will still be needing three more correct tosses of coin to complete our respective sequences. If we get heads, then we continue.
Step 2: If either of us tosses the coin to get heads, we stay on this very step. If we get tails, we continue.
Step 3a (me): If I tosses the coin to get tails, I will have to go back to the step one again and try till I get heads again. Thus I will need at least 3 more tosses to complete my sequence.
Step 3b (my friend): If my friend tosses the coin to get heads, he will go back to step 2 again and will try till he gets tails. In this scenario, he will need at least 2 more tosses to complete the sequence.
Clearly, he has the better opportunity to win this game.