My friend and i usually flip a quarter to see who buys lunch. Since i won the last three lunches in a row i told my friend we would do the next coin flip as follows to give him an advantage. I would like to know if he really does have an advantage and if so what % advantage does he have. Here is the play. My friend takes heads and I tails. We continue to flip the quarter until either my friend gets ahead of me by 2 or i get ahead of him by three. This could take several flips of the quarter until he has an advantage of 2 over me or i have gained an advantage of 3 over him. Is there someone out there that could confirm that my friend has an advantage and what % that would be?
probability? (flipping a coin to see who pays for lunch)
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pka
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I do not understand what that phrase could mean. Please clarify its meaning with an exact example.
I will suggest that you consider the following.
Your friend flips first and you next alternating until he has two heads or you have three tails.
Note that means he could win on or after his second try but you can win on or after your third try.
Thank you very much for your message. Here is an example. Note, My friend likes to flip coins so I let him do all the coin flipping.
1st flip: Heads. my friend has gained an advantage of 1
2nd flip: Tails, my friend has lost his advantage of 1, we are now back to even or if you will tied just as we were when we began.
3rd flip: Tails, I have gained an advantage of 1
4th flip: Tails, I have gained an advantage of 2
5th flip: Heads, I now have an advantage of 1
6th flip: Tails, I now have an advantage of 2
7th Flip: Tails, I now have gained an advantage of 3 over my friend before my friend gained an advantage of 2 over me. He must now pay for lunch.
It would seem to me that only having to achieve an advantage of 2 instead of an advantage of 3 would have a higher probability of occurring. If that is true then what is his advantage over me as a percentage? i.e. in 100 games is he more likely to win 60 compared to me winning 40?
I would very appreciate your expert analysis.
1st flip: Heads. my friend has gained an advantage of 1
2nd flip: Tails, my friend has lost his advantage of 1, we are now back to even or if you will tied just as we were when we began.
3rd flip: Tails, I have gained an advantage of 1
4th flip: Tails, I have gained an advantage of 2
5th flip: Heads, I now have an advantage of 1
6th flip: Tails, I now have an advantage of 2
7th Flip: Tails, I now have gained an advantage of 3 over my friend before my friend gained an advantage of 2 over me. He must now pay for lunch.
It would seem to me that only having to achieve an advantage of 2 instead of an advantage of 3 would have a higher probability of occurring. If that is true then what is his advantage over me as a percentage? i.e. in 100 games is he more likely to win 60 compared to me winning 40?
I would very appreciate your expert analysis.
pka
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Thank you for the detail example. I had not thought of your loosing advantage.
The bad news is that this makes a very difficult question.
Whole textbooks have been written on random walks. Click on that link and you will see.
This is a subarea of Markov Chains. The transition matrix is here.
That calculation is for five flips. In the result, he third row are the probabilities that starting at 0 one would end up at -2 -1 0 1 2 or 3. However the downside it does tell you if one of you has been there before.
This is a very advanced question. Sorry but there is no quick answer.
First of all my friend, i would like to express my sincerest appreciation for the time and energy you have given of yourself to answer my question. My math teachings ended 50 years ago in high school and topped out at pre-algebra i'm embarrassed to say. As a result the information you have provided is major heavy. That said, it has given me an incentive to pursue an answer if there is one, to my question. I'm anixous to inform my friend of the math complexities our 'simple' game to decide who pays for lunch has presented. Many thanks!
This is getting booooring...
why don't you guys play X and O instead, and you let your friend start;
if he keeps losing given the start, then he deserves to pay lunch
Sorry about that. It was not intended. I do want thank you for your alternative solution for determining which one of us pays for lunch. My friend is a die hard coin flipper. Comes from back in the day when coin flipping was very popular and the preferred method by most to provide an answer for a broad range of things that could not be agreed upon by the parties involved. Thank you very much!