4 ms·

The problem is not well-posed. The debate should be about its formulation, not about the math... The key here is to understand that no matter what you do, the

by ucha 12y ago

The problem is not well-posed. The debate should be about its formulation, not about the math... The key here is to understand that no matter what you do, the game organizer will ALWAYS open one of the two others doors. Now, what if the organizer was to open a door only in some cases? He could open the door only when you choose the correct door. In that case, you have 0 chance of winning when you switch doors when the organizer opens another one. What makes the 1/2-1/2 probability - that so may math PhDs defended - acceptable is this: the organizer is selecting a door at random and opening it. Then if I choose a door and the organizer opens another one, I know that I have a 50% chance of having selected the correct one. Of course, he could also have opened my door in which case the game ends. This is a good example of how an ill-posed problem have different solutions depending on its implicit interpretation.

mcphage 12y ago
In the problem as stated, the organizer did open a door, which did contain a goat.
- dllthomas 12y ago
  How the organizer chose the door matters.
  - mcphage 12y ago
    If you didn't know how the organizer chose the door (which you do, the original problem clearly states it), then your decision needs to be made independently of how the organizer chose it. All you can use are the facts in front of you: you chose a door, the organizer chose a different door, that contained a goat, and offered you the chance to choose. You need to make the best decision you can with the facts that you have.
    - dllthomas 12y ago
      My point was that your listing of things that happened in the past omitted an important piece of it. I actually don't agree that "the original problem clearly states it." Fully cognizent of the issues at hand, and with space for reflection entirely outside of the context of trying to solve the problem, I agree that it is clear that this is what the statement of the problem was trying to get at (particularly with the note that Monty knows which door has what) but it strictly does not say that that informs his choice (or how).
- ucha 12y ago
  That's not enough. You need the knowledge of "the organizer will ALWAYS open a door that contains a goat" to draw the conclusion of the article. That knowledge is implicitly assumed. I have shown that by assuming a different knowledge - in this case, the organizer only opens the door that contains a goat when you choose the door that contains the car - you can end up with different probabilities. I don't have an IQ of 200+ nor do I have a PhD in math, but I'm pretty certain that the probabilities in this problem are unknown unless you know the behavior of the organizer in EVERY situation, not in one particular case.
  - mcphage 12y ago
    > That's not enough. You need the knowledge of "the organizer will ALWAYS open a door that contains a goat" to draw the conclusion of the article. That knowledge is implicitly assumed. It's not implicitly assumed, it's stated in the original question: "You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat." You pick a door, the hosts opens another door which contains a goat. He can't open the door you chose, he can't open a door with the prize, he can't not open a door. Introducing a possibility which is not a part of the specified problem turns it into a different problem. > I have shown that by assuming a different knowledge - in this case, the organizer only opens the door that contains a goat when you choose the door that contains the car - you can end up with different probabilities. For every "the organizer only the opens a door that contains a goat when you choose the door that contains a car", there's an "the organizer only opens a door that contains a goat when you open a door that doesn't contain a car". Not knowing the host's behavior means that it isn't part of your analysis. If you don't know why he does what he does, all you can use are the facts in front of you: there are three doors, you pick one, you are shown a second door which doesn't contain the prize, and are offered a chance to choose. > I'm pretty certain that the probabilities in this problem are unknown unless you know the behavior of the organizer in EVERY situation, not in one particular case Not so. If you knew the host's behavior (which the original problem does specify precisely), you can use it in your analysis. But not knowing it, means the best decision has to be independent of it. And if you knew the door that contained the prize, you could use that in your analysis. But not knowing it, means the best decision has to be made independent of it.

by ucha 12y ago

mcphage 12y ago

In the problem as stated, the organizer did open a door, which did contain a goat.

dllthomas 12y ago
How the organizer chose the door matters.
- mcphage 12y ago
  If you didn't know how the organizer chose the door (which you do, the original problem clearly states it), then your decision needs to be made independently of how the organizer chose it. All you can use are the facts in front of you: you chose a door, the organizer chose a different door, that contained a goat, and offered you the chance to choose. You need to make the best decision you can with the facts that you have.
  - dllthomas 12y ago
    My point was that your listing of things that happened in the past omitted an important piece of it. I actually don't agree that "the original problem clearly states it." Fully cognizent of the issues at hand, and with space for reflection entirely outside of the context of trying to solve the problem, I agree that it is clear that this is what the statement of the problem was trying to get at (particularly with the note that Monty knows which door has what) but it strictly does not say that that informs his choice (or how).
ucha 12y ago
That's not enough. You need the knowledge of "the organizer will ALWAYS open a door that contains a goat" to draw the conclusion of the article. That knowledge is implicitly assumed. I have shown that by assuming a different knowledge - in this case, the organizer only opens the door that contains a goat when you choose the door that contains the car - you can end up with different probabilities. I don't have an IQ of 200+ nor do I have a PhD in math, but I'm pretty certain that the probabilities in this problem are unknown unless you know the behavior of the organizer in EVERY situation, not in one particular case.
- mcphage 12y ago
  > That's not enough. You need the knowledge of "the organizer will ALWAYS open a door that contains a goat" to draw the conclusion of the article. That knowledge is implicitly assumed. It's not implicitly assumed, it's stated in the original question: "You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat." You pick a door, the hosts opens another door which contains a goat. He can't open the door you chose, he can't open a door with the prize, he can't not open a door. Introducing a possibility which is not a part of the specified problem turns it into a different problem. > I have shown that by assuming a different knowledge - in this case, the organizer only opens the door that contains a goat when you choose the door that contains the car - you can end up with different probabilities. For every "the organizer only the opens a door that contains a goat when you choose the door that contains a car", there's an "the organizer only opens a door that contains a goat when you open a door that doesn't contain a car". Not knowing the host's behavior means that it isn't part of your analysis. If you don't know why he does what he does, all you can use are the facts in front of you: there are three doors, you pick one, you are shown a second door which doesn't contain the prize, and are offered a chance to choose. > I'm pretty certain that the probabilities in this problem are unknown unless you know the behavior of the organizer in EVERY situation, not in one particular case Not so. If you knew the host's behavior (which the original problem does specify precisely), you can use it in your analysis. But not knowing it, means the best decision has to be independent of it. And if you knew the door that contained the prize, you could use that in your analysis. But not knowing it, means the best decision has to be made independent of it.