I was playing heads or tails...

T

Thales12345

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Imagine that you are playing heads or tails 5 times with a group of x people. At least half of the group guesses correctly in every single game.

It strikes me that you can always find 2 people so that in each separate game at least one of them guesses correctly. How is this possible? Can this be proven?
 
Thales12345 said:
Imagine that you are playing heads or tails 5 times with a group of x people. At least half of the group guesses correctly in every single game.

It strikes me that you can always find 2 people so that in each separate game at least one of them guesses correctly. How is this possible? Can this be proven?
Click to expand...
Don't understand the question. But any 2 people can decide to always guess one heads and the other tails, so exactly one of them will guess correctly every time.
 
Yes of course, but I mean if they don't decide ahead of time. Think of it as a test with 5 questions, which are corrected as correct or incorrect. I'm amazed that if every question is answered correctly by at least half of the students, you can always find 2 students so that every question is answered correctly by at least 1 of those 2.
 
Thales12345 said:
Imagine that you are playing heads or tails 5 times with a group of x people. At least half of the group guesses correctly in every single game.

It strikes me that you can always find 2 people so that in each separate game at least one of them guesses correctly. How is this possible? Can this be proven?
Click to expand...
How is what possible? You assert a hypothesis and then ask why it is true. Perhaps it is not true.

Is a “game” guessing correctly the outcomes of five distinct tosses of a fair coin? Why do you believe that if four people were writing down the outcomes, at least two people would guess all five outcomes correctly?
 
By the way, you have not specified whether they are to write in advance the number of heads and tails or the sequence of heads and tails.

This is a very vague question.
 
It is given that every question is answered correctly by at least half of the students. Whenever I apply this to fictional students A, B, C, D... I can always find 2 students so that each of these 5 questions is answered correctly by at least 1 of those 2. I was wondering if this is ALWAYS the case . If so, can this be proved in mathematics?

In the example of heads or tails, they predict the outcome per game. When at least half of the group always predicts the outcome correctly, I can always find 2 people so that every game is correctly predicted by at least 1 of them.

Hopefully it’s clear now...
 
It is given by WHOM? You seem to be reading something that you do not quote.

If what you are reading about is FREQUENCY statistics related to a specific test taken by specific students, students do not answer at random. You cannot use flipping a coin for analyzing students answering a test: randomness applies in once case but not the other..
 
Thales12345 said:
Imagine that you are playing heads or tails 5 times with a group of x people. At least half of the group guesses correctly in every single game.

It strikes me that you can always find 2 people so that in each separate game at least one of them guesses correctly. How is this possible? Can this be proven?
Click to expand...
Thales12345 said:
Think of it as a test with 5 questions, which are corrected as correct or incorrect. I'm amazed that if every question is answered correctly by at least half of the students, you can always find 2 students so that every question is answered correctly by at least 1 of those 2.
Click to expand...
Thales12345 said:
It is given that every question is answered correctly by at least half of the students. Whenever I apply this to fictional students A, B, C, D... I can always find 2 students so that each of these 5 questions is answered correctly by at least 1 of those 2. I was wondering if this is ALWAYS the case . If so, can this be proved in mathematics?

In the example of heads or tails, they predict the outcome per game. When at least half of the group always predicts the outcome correctly, I can always find 2 people so that every game is correctly predicted by at least 1 of them.
Click to expand...

My first thought was that students are not coins. They don't answer questions randomly.

On the other hand, your question is not really about probability in the first place, but, now that I think about it, sounds more like a pigeonhole principle question.

Here is what I think you are saying:

SUPPOSE that a 5-question T/F test has been taken by N people, AND for each question, at least half the students got the correct answer.​
THEN, can it be proved that there MUST be some one pair of students such that at least one of them got each answer right?​

So this is not about probabilities or randomness, as initially suggested by the coins and your putting it under probability; and it supposes something that you are not saying always happens, but that appears to imply your conclusion.

I think also that this is not a problem you were assigned, but a question that came to your mind as a result of observation.

Is that a correct interpretation of your question?
 
Dr.Peterson said:
THEN, can it be proved that there MUST be some one pair of students such that at least one of them got each answer right?​

I think also that this is not a problem you were assigned, but a question that came to your mind as a result of observation.

Is that a correct interpretation of your question?
Click to expand...
If each question separately is answered correctly by at least half of the students, I think you can find 2 people with the following requirement:
For each question at least 1 of these 2 must answer the question correctly. This MUST not always be the same person. A question can also be answered correctly by both people.
(for example:
question 1 was answered correctly by person A,
question 2 was answered correctly by person B,
question 3 was answered correctly by person A and B,
question 4 was answered correctly by person B
and question 5 was answered correctly by person A and B)

What do you think? Can you always find such a pair and is it provable?

(And yes, I was thinking about this myself. It's been a while since I was assigned tasks at school :))
 
I believe you are saying that I interpreted the problem correctly.

To make things simple, we can suppose that the correct answer for each question is T.

I mentioned the pigeonhole principle, which you can look up. It's often used in fancy ways, but often I just think this way: If your goal is to find such a pair every time, my goal as your nemesis is to prevent you from doing so, by arranging the answers so it can't happen. That is, I want to find a counterexample to your claim (a set of answers to the 5 questions by N people, for which no pair got a T for each question). I'm hoping to find something that will prevent me from doing so.

It's easy to show that I can't find a counterexample for 3 people. If each row is a question and each column is a person, each row has to have at least two T's in it, because at least half must be correct. I'll try to make sure that every pair of columns has FF in some row. But I can't put FF in any row! There can only be one F. So it has to look something like this at worst:

FTT​
TFT​
TTF​
FTT​
TFT​

Here every pair fits your requirement; for example, the first and last columns taken alone are

FT​
TT​
TF​
FT​
TT​

and at least one of them got T in every row.

Try doing this with 4 people, and see what happens. If you can find a counterexample, you've proved your claim wrong; if you can't, try to see why you couldn't, and that may be the proof you're asking for.
 
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