- Joined
- 26 Apr 03
- Moves
- 26771
Hmmmm......
Sorry, I don't think your conclusion is correct.
Originally posted by Acolyte
Assuming the guard uniformly picks one of the condemned (possibly including A), the guard's responses are symmetric, so we can consider what happens if the guard says 'B and C', and we'll call this event D. Let A, B and C be the events that the respective prisoner is to be spared. Then
P(A|D) = P(AnD)/P(D) = P(A)*P(D|A)/P(D)
P(A) = P(D) = 1/3
P(D|A) = 1/2 (the guard chooses one of B and C, then chooses the other with prob. 1/2)
=> P(A|D) = 1/2
By symmetry P(B|D) = 1/4, P(C|D) = 1/4
So the guard is guaranteed to give information to all three prisoners about their fate, with the obvious result that the prisoner not named is more likely to be spared.
Originally posted by lilmissnaughtyMaybe that's because people post all sorts of 'confusing' variants before agreeing on the answer to the original.
all of this is confusing me I will tell you what, I will jump into the prison and shoot them all so they all have the same probability of dying! problem solved!
TheMaster37 may like to read the 3rd paragraph of the following page: http://mathworld.wolfram.com/PrisonersDilemma.html
.
Originally posted by CribsI still don't see how the prisoners are given a choice, something that is crucial in the monty hall problem...
Substitute "Live" for "Prize", "Die" for "No Prize", and "Would you rather be Prisoner C?" for "Would you like to switch to the remaining closed door?". Further, specify that the guard knows from the outset everybody's fate, just as Monty knows the contents of every door. Prisoner A asking to reveal the death of B or C is analogous to the contes ...[text shortened]... Prisoner C were the one playing the game, he would end up wishing he were Prisoner A.)
Cribs
Originally posted by TheMaster37After hearing that B will die [there is no prize behind this door], the choice for A is, "Would you like to swap places with C?" ["Would you like to change doors?"]
I still don't see how the prisoners are given a choice, something that is crucial in the monty hall problem...
According to you, it makes no difference.
According to me, he doubles his chances of surviving by doing so.
.
Originally posted by TheMaster37Suppose in the Monty Hall problem, the contestent didn't actually have the option to switch. It still doesn't change the fact that it would be better for him to switch. The same applies in the prisoner problem. A doesn't have the option to suddenly become C, but it would be better for him if he could.
I still don't see how the prisoners are given a choice, something that is crucial in the monty hall problem...
In other words, contrary to what you say, it is NOT crucial that the contestent
has the option to switch in the Monty Hall problem; in fact it is completely
irrelevant.
Cribs
To revamp an old post
Ok, I have absolutely no idea what teh answer is but here is something to think about. This may make no sense and be completely wrong but here it is anyway. Before A asks the question A has 1/3 chance od survival, B has 1/3 and C has 1/3 making a total of 1 when added together. Now A asks the question, gets an answer and now has 1/2 chance. What if B and C dont hear the answer and then ask EXACTLY the same question to the guard? THey will each think that they have a 1/2 chance of survival. Add up the probabilities to give a total of 1.5 As I said, any or all of this could be crap but just a thought i had at 2:30 am in the morning
Originally posted by JezzThe Law of Total Probability only holds when everybody has identical information about the uncertain event.
To revamp an old post
Ok, I have absolutely no idea what teh answer is but here is something to think about. This may make no sense and be completely wrong but here it is anyway. Before A asks the question A has 1/3 chance od survival, B ...[text shortened]... s could be crap but just a thought i had at 2:30 am in the morning
Consider this analogy. We shuffle a deck of cards. You peek at the top card, and I don't. You see that the top card is red, but don't tell me.
Now, I correctly conclude that the top card is black with probability 1/2. You correctly conclude that the top card is red with probability 1, because you have different information. The "total probability," as you would have it, is 1.5, but this is a mistaken application of summing probabilities, for each is based on different conditional information.
To return to the original problem, since B doesn't know the answer to A's question, B and A have different information, and thus are entitled to be free of the Law of Total Probability when making their conclusions. Throw C and his own information into the mix, and that freedom still exists. The fact that the event probabilities don't sum to 1 across an information gap does not make their conclusions incorrect.
Dr. S
Originally posted by JezzAs an alternate refutation to your analysis, consider that to reach the state in which each of A, B, and C correctly conclude that they have a 1/2 chance of survival, the guard had to have lied to one of them. That is, he told A that, with certainty, either B or C will be killed. That means that the guard is not allowed to both tell B that A or C will definitely be killed and tell C that A or B will definitely be killed, provided that it is implicit in the problem formulation that the guard is not allowed to lie. (If he is, then we can't analyze the problem at all.)
To revamp an old post
Ok, I have absolutely no idea what teh answer is but here is something to think about. This may make no sense and be completely wrong but here it is anyway. Before A asks the question A has 1/3 chance od survival, B ...[text shortened]... s could be crap but just a thought i had at 2:30 am in the morning
So, your 1.5 paradox can never be attained, and thus does not refute the correct solution.
Thats cool. I knew my logic must have been wrong I just could not see where. However, the guard need not lie. If B and C are to be killed, the guard can tell A that B is certain to die, he can tell B that C is certain to die and he can tell C that B is certain to die. Thus they all think that they have a 50% of surviving and the guard did not lie.
The Master has got it right. There is no "switch" variant, so it's not a variant of the Monty Hall problem.
In the Monty Hall problem there is a choice after the new information.
Here, there is none, so his chances are now 50% since the guard has eliminated the possibility of A&C both being executed.
If he could switch cells with C and the chances of survival depended on which cell he was, then he should switch. But that was not stated in the original problem.
Originally posted by PalynkaOh my God, must we go through this again, when I have already shown several posts above that having a choice is completely irrelevant. The Monty Hall analysis would not be affected in the slightest if the contestant didn't actually get to choose; it remains true that it would be correct for him to, even if Monty didn't allow it.
The Master has got it right. There is no "switch" variant, so it's not a variant of the Monty Hall problem.
In the Monty Hall problem there is a choice after the new information.
Here, there is none, so his chances are now 50% sin ...[text shortened]... should switch. But that was not stated in the original problem.
A perfect analogy between all of the components of the two problems has already been drawn. They differ only in their terminology; the correct analysis is identical.
The reasoning that leads you to your 50% solution is flawed. The 1/3 solution is correct under your interpretation of the problem. I have shown an alternate and less-favorable interpretation of the problem which has 50% as a valid solution, but your analysis does not address that version.
Originally posted by THUDandBLUNDERMy brain hurts😞
There are 3 prisoners in a prison. Let’s call them A, B, and C. Tomorrow, 2 of them will be executed, but the prisoners don’t know which of them have been chosen. Prisoner A reasons that his chances of survival are 1/3 (a third). He ...[text shortened]... nces of survival have increased from 1/3 to 1/2. Is he right?
.