Well, "No More!" he cried, in his best James Belushi voice.
No more standing out in the courtyard!
No more looking at everyone else's hat!
No more pieces of paper and pencils held in the right hand or the left hand!
No more standing facing this way or that way.
No more turning of the head to the left or the right!
No More!
Gee, he said to himself, I feel much better now.
He has just visited his men this evening and told them how it will be, in the morning.
First:
In the morning he will visit a cell at random and lead a prisoner to his office and seat him where no one else can see him or hear him. None of the other prisoners will know who that prisoner is. He will then visit another cell of his choosing and lead that prisoner to a solitary confinement cell, where he can not be seen or heard; neither can he see nor hear anything or anyone else. None of the other prisoners will know who that prisoner is. As they are led from their cells, a hat will be placed on each of their heads - randomly for each - either a black hat or a white hat. Neither of them will know the color of his own hat.
Next:
The warden will visit each prisoner in turn and ask him what color his hat is. As a clue, the warden will tell him the color of the other prisoner's hat. But he will not tell him the other prisoner's name. Naturally, the warden will not wink, nod, turn left or right, stand forward or backward, or in any way signal any extra information to the prisoner.
Next:
The warden will go to the courtyard and decide what to do with the two prisoners. He will decide as follows: If both prisoners have guessed right, they both will be executed! If both prisoners have guessed wrong, they both will be executed! However, if one prisoner has guessed right and the other prisoner has guessed wrong, they both will be set free. The warden will bring the good [or bad] news to the pair of prisoners, and, out of the sight and hearing of all the others, they will meet their fate: either the gallows, or a cab ride to the nearest Denny's for a decent breakfast and $50 to find their way home.
This process will be repeated until all 100 prisoners' fates have been decided.
Once again, you have been called in to advise the prisoners on an optimal strategy for tomorrow's test. How do you advise them, and how many do you expect to save?
"Expect" in the statistical sense means, if you repeat the experiment an arbitrarily large number of times, the average outcome is what you would "expect" for a single outcome. For example the expectation for two coin tosses is 1 head and 1 tail - based on the average results of a large number of tosses.
Just to make it clear: The warden will choose the "pairs" of prisoners tomorrow. The prisoners cannot advantageously pair themselves up tonight. Nor will the warden reveal the name of the other prisoner in each pair, when each is asked his color.
Question
bonanova
Warden Jones was infuriated to learn how the scheming, conniving, diabolically devious Brainden-ers advised his 100 prisoners the last time he decided to give them a chance at their freedom.
Well, "No More!" he cried, in his best James Belushi voice.
No more standing out in the courtyard!
No more looking at everyone else's hat!
No more pieces of paper and pencils held in the right hand or the left hand!
No more standing facing this way or that way.
No more turning of the head to the left or the right!
No More!
Gee, he said to himself, I feel much better now.
He has just visited his men this evening and told them how it will be, in the morning.
First:
In the morning he will visit a cell at random and lead a prisoner to his office and seat him where no one else can see him or hear him. None of the other prisoners will know who that prisoner is. He will then visit another cell of his choosing and lead that prisoner to a solitary confinement cell, where he can not be seen or heard; neither can he see nor hear anything or anyone else. None of the other prisoners will know who that prisoner is. As they are led from their cells, a hat will be placed on each of their heads - randomly for each - either a black hat or a white hat. Neither of them will know the color of his own hat.
Next:
The warden will visit each prisoner in turn and ask him what color his hat is. As a clue, the warden will tell him the color of the other prisoner's hat. But he will not tell him the other prisoner's name. Naturally, the warden will not wink, nod, turn left or right, stand forward or backward, or in any way signal any extra information to the prisoner.
Next:
The warden will go to the courtyard and decide what to do with the two prisoners. He will decide as follows: If both prisoners have guessed right, they both will be executed! If both prisoners have guessed wrong, they both will be executed! However, if one prisoner has guessed right and the other prisoner has guessed wrong, they both will be set free. The warden will bring the good [or bad] news to the pair of prisoners, and, out of the sight and hearing of all the others, they will meet their fate: either the gallows, or a cab ride to the nearest Denny's for a decent breakfast and $50 to find their way home.
This process will be repeated until all 100 prisoners' fates have been decided.
Once again, you have been called in to advise the prisoners on an optimal strategy for tomorrow's test. How do you advise them, and how many do you expect to save?
"Expect" in the statistical sense means, if you repeat the experiment an arbitrarily large number of times, the average outcome is what you would "expect" for a single outcome. For example the expectation for two coin tosses is 1 head and 1 tail - based on the average results of a large number of tosses.
Just to make it clear: The warden will choose the "pairs" of prisoners tomorrow. The prisoners cannot advantageously pair themselves up tonight. Nor will the warden reveal the name of the other prisoner in each pair, when each is asked his color.
Think well - lives hang in the balance.
Most of all ... have fun
After all, you won't be executed...!
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