Hint: 1-player agree on system to choose a strategy based on what they see. 2- t
ID: 3071343 • Letter: H
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Hint: 1-player agree on system to choose a strategy based on what they see.
2- two total color solections are equal good(both win or both lose) iff they differ at only finitely many people.
3- any given player guess for correct total color selection will win.
7. A set of people will have hats placed on their heads so that they can each see all of the other hats but not their own. There are two hat colors: red and blue. Each person has one chance to guess the color of their own hat. They all win if only a finite number of them guess incorrectly. They understand how the game will go and can plan a strategy in advance. Prove that assuming the Axiom of Choice there is a winning strategy targ.Explanation / Answer
GIVEN - There are hats of two colour red and blue. They can see each others hat but not of themselves.The probability of choosing red or blue hat is same i.e. there are equal number of red and blue hats.
TO PROVE - We have to prove that the axiom of choice they are assuming will make them win.
PROOF - as we know that they can see each other's hat. So the strategy is that firstly they will stand in a line with alternate colour of hats. As people can see each others hat so they can help each other in making such a line.
If the 1st person gusses the colour of his/her hat to be Blue/Red then the next person beside him will know his hat's colour, which will be different from the 1st person's hat.
For example, if the 1st person's hat is Red then the 2nd person will know that his hat's colour is Blue and so on the 3rd person will know that his hat's colour is Red and this will continue.
At last, if the given person gusses the colour of his Hat right, then they will surely win the game.
This proves that, the Axiom of choice they were assuming will make them win the game.
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