The Prisoners’ Paradox
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The Prisoners’ Paradox
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Re: The Prisoners’ Paradox
#2Re: The Prisoners’ Paradox
#3http://en.wikipedia.org/wiki/Monty_Hall_problem
Yeah, except it's pretty obvious why there is a switch to 1/2 from 1/3 in Prisoner's Paradox and it's not obvious in MH.
What I understand happened in Prisoner's Paradox is that the very field of probabilistic distribution of pardons has shrinked from 3 to 2. It's like binary search. You cheated your way by knowing where _not to look_.
Another simple analogy: you can increase the value of rational number both by increasing the numerator and decreasing denominator. In this case you decreased denominator.
Monty Hall requires a slightly bigger leap of logic.
Actually, here's a more directly linked problem: http://en.wikipedia.org/wiki/Three_Prisoners_problem
Re: The Prisoners’ Paradox
#4http://en.wikipedia.org/wiki/Monty_Hall_problem
Yeah, except it's pretty obvious why there is a switch to 1/2 from 1/3 in Prisoner's Paradox and it's not obvious in MH. What I understand happened in Prisoner's Paradox is that the very field of probabilistic distribution of pardons has shrinked from 3 to 2. It's like binary search. You cheated your way by knowing where _not to look_. Another simple analogy: you can increase the value of rational number both by incr…
That means that at least one of my cellmates is still sure to die. Give me his name
Prisoner A rejoices that his own chance of survival has improved from 1/3 to 1/2. But how is this possible? The guard has given him no new information. Has he?
No, Prisoner A still has a 1 in 3 chance of [edit: not] dieing. If he asked if Prisoner B had a pardon then you could argue his odds became 1/2. But, now he still only knows that at least one of his cellmates is going to die.
Re: The Prisoners’ Paradox
#5As non-intuitive puzzles go I think the hat guessing game (http://www.relisoft.com/science/hats.html) takes the cake. I still remember a dailywtf thread where 19 pages worth of people claim that the correct solution is completely bogus (http://thedailywtf.com/Comments/Riddle-Me-An-Interview.aspx).
Re: The Prisoners’ Paradox
#6Earlier quoted context omitted.
Yeah, except it's pretty obvious why there is a switch to 1/2 from 1/3 in Prisoner's Paradox and it's not obvious in MH. What I understand happened in Prisoner's Paradox is that the very field of probabilistic distribution of pardons has shrinked from 3 to 2. It's like binary search. You cheated your way by knowing where _not to look_. Another simple analogy: you can increase the value of rational number both by incr…
That means that at least one of my cellmates is still sure to die. Give me his name Prisoner A rejoices that his own chance of survival has improved from 1/3 to 1/2. But how is this possible? The guard has given him no new information. Has he? No, Prisoner A still has a 1 in 3 chance of [edit: not] dieing. If he asked if Prisoner B had a pardon then you could argue his odds became 1/2. But, now he still only knows th…
I think you've got it backwards. Only one is pardoned. His chance of survival is 1 in 3.
Re: The Prisoners’ Paradox
#7A's chances has not changed.
But the chance of C being the pardoned has doubled to 2/3.
Re: The Prisoners’ Paradox
#8Earlier quoted context omitted.
That means that at least one of my cellmates is still sure to die. Give me his name Prisoner A rejoices that his own chance of survival has improved from 1/3 to 1/2. But how is this possible? The guard has given him no new information. Has he? No, Prisoner A still has a 1 in 3 chance of [edit: not] dieing. If he asked if Prisoner B had a pardon then you could argue his odds became 1/2. But, now he still only knows th…
I think you've got it backwards. Only one is pardoned. His chance of survival is 1 in 3.
The initial chance of survival, after hearing that there is a chance, (for Prisoner A) is 1/3, since any one prisoner dies out of three. If he now knows, after consulting with the guard, that Prisoner B is sure to die that means that it's death for either A or C. Hence the chance of survival (and death since P(Death) = 1 - P(Survival)) is now 0.5. This results in a smaller chance of death (for him) 1/2 rather than 2/3. Hence the rejoicing.
Re: The Prisoners’ Paradox
#9Oh dear. So this is an implementation of the Monty Hall problem but now is crossing over into Metaphysics.
There are several different ways to consider the ontological reality of probabilities (i guess wikipedia only has two listed, Frequentists and Bayesians http://en.wikipedia.org/wiki/Probability_interpretations ).
The thought experiment in question probably should be answered with a Bayesian world view, and does not appear to be.
Prisoner A's chances of being executed is not materially effected when he learns new information, because the decision as to who has been pardoned has already been made.
Finding out new information, does change a probability, but not the probability that he'll be executed. It only changes the likelihood of whether or not the assertion that he will be executed is true. The nature of the universe has not actually changed, merely the state of belief of Prisoner A.
The point being, he's got no cause to celebrate, upon merely discovering information, because learning more information does not actually change the state of the world, in this case.
Re: The Prisoners’ Paradox
#10http://en.wikipedia.org/wiki/Monty_Hall_problem
Yeah, except it's pretty obvious why there is a switch to 1/2 from 1/3 in Prisoner's Paradox and it's not obvious in MH. What I understand happened in Prisoner's Paradox is that the very field of probabilistic distribution of pardons has shrinked from 3 to 2. It's like binary search. You cheated your way by knowing where _not to look_. Another simple analogy: you can increase the value of rational number both by incr…
You mean it's pretty obvious why there is no switch to 1/2 from 1/3? As the Wikipedia article you linked points out: "The answer is he didn't gain information about his own fate, but he should switch with C if he can."
This isn't the same as Monty Hall. Monty Hall only increases the odds after switching doors. This actually seems like a trick meant to fool people who know about the Monty Hall problem but don't really understand it.