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3 Door Logic Problem

Roboramma said:
Actually, human nature being what it is, I think the opposite is true. If you play for money, particularly with a stranger, you can expect them to take the best strategy possible. And if the rules don't prohibit it, making you the offer only when it's in their favour is the best strategy.
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:) Ok, play the game with a wad of paper or rubber ball and three cups. The money is entirely beside the point.
 
RandFan said:
You are making unwarranted assumptions that there will be some sort of trickery.
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Why is the assumption that there will be some sort of trickery more unwarranted than the assumption that there will not be some sort of trickery? The point is, we don't _know_ how the host comes to the decision to offer us a chance to switch. He could be doing his utmost to help us (only asking us to switch when we've made the wrong initial choice) or his utmost to hurt us (only asking us to switch when we've made the right initial choice) or anything in between.
 
Number Six said:
Why is the assumption that there will be some sort of trickery more unwarranted than the assumption that there will not be some sort of trickery? The point is, we don't _know_ how the host comes to the decision to offer us a chance to switch. He could be doing his utmost to help us (only asking us to switch when we've made the wrong initial choice) or his utmost to hurt us (only asking us to switch when we've made the right initial choice) or anything in between.
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You'll have to give me a reason to believe that the host would want to trick people. In the TV show it was for dramatic effect. The average person has no such reason. I would have to assume the law of averages or otherwise have reason to suspect trickery. As I have said, I have played the game with a number of people and not one of them thought to try and change the game.

If you could demonstrate that it would occur to the average person to want to make a change and that the average person would then make the change then I would change my mind. As it is we don't have enough information to assume that the host is somehow different than the average person and therefore would be as likely to make the change as not. That is why I say that you are making unwarranted assumptions. Remember, a choice of two outcomes is not necessarily 50/50.

I suppose you could argue that the average person doesn't make such an offer and that a person who did make such an offer would likely have motivations different from the average person. Perhaps but again you would be making assumptions not presented in the original hypothetical. Which by the way is just a hypothetical that doesn't call for additional premises or contemplation of such premises. If you wish to refine the hypothetical that is fine but as it is it doesn't call for such assumptions. IMO.

I'd switch.
 
RandFan, it's you who are making unwarranted assumptions. Everyone else here is saying that without making assumptions about the host's motivations, there is not enough information to solve the problem. The way you are solving it, and arriving at the 2/3 answer, is by making an assumption that's not described in the puzzle statement, namely that the host is constrained to always offer the switch. Without some assumption about his methods, the puzzle can't be solved. This isn't "changing" the puzzle, it's sticking to its wording. If anything, making an unstated assumption is changing it.
But you are adding to the game variables that are not likely. If I ask 100 people to play host how many of them will do as you would?
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Voting isn't a very good technique for solving probability problems.
 
Folks, the correctly stated "Monty Hall Problem" makes it clear that Monty knows what's behind the doors and shows you one with a goat:
A thoroughly honest game-show host has placed a car behind one of three doors. There is a goat behind each of the other doors. You have no prior knowledge that allows you to distinguish among the doors. "First you point toward a door," he says. "Then I'll open one of the other doors to reveal a goat. After I've shown you the goat, you make your final choice whether to stick with your initial choice of doors, or to switch to the remaining door. You win whatever is behind the door."
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~~ Paul
 
CurtC said:
RandFan, it's you who are making unwarranted assumptions. Everyone else here is saying that without making assumptions about the host's motivations, there is not enough information to solve the problem. The way you are solving it, and arriving at the 2/3 answer, is by making an assumption that's not described in the puzzle statement, namely that the host is constrained to always offer the switch. Without some assumption about his methods, the puzzle can't be solved. This isn't "changing" the puzzle, it's sticking to its wording. If anything, making an unstated assumption is changing it.
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In fact, after getting beat over the head enough times, Marilyn did acknowledge the fact that, yes, it needed this assumption. However, she tried to get out of it by claiming it was an obvious and minor assumption to make.

I say, fine, call it what you want, but admit that you need to make the assumption in order to solve the problem, because it is not explicitly specified in the statement of the problem.
 
The version Paul A used is:

A thoroughly honest game-show host has placed a car behind one of three doors. There is a goat behind each of the other doors. You have no prior knowledge that allows you to distinguish among the doors. "First you point toward a door," he says. "Then I'll open one of the other doors to reveal a goat. After I've shown you the goat, you make your final choice whether to stick with your initial choice of doors, or to switch to the remaining door. You win whatever is behind the door."

This version makes it clear that the host is telling you _beforehand_ that he is going ask you if you want to switch. That makes all the difference. Most or all versions of this I've seen didn't do that. Also, this version doesn't say that you know that there is a car behind one door and a goat behind each of the other two, although just about every other version I've seen does.
 
RandFan said:
You'll have to give me a reason to believe that the host would want to trick people. In the TV show it was for dramatic effect. The average person has no such reason. I would have to assume the law of averages or otherwise have reason to suspect trickery. As I have said, I have played the game with a number of people and not one of them thought to try and change the game.

If you could demonstrate that it would occur to the average person to want to make a change and that the average person would then make the change then I would change my mind. As it is we don't have enough information to assume that the host is somehow different than the average person and therefore would be as likely to make the change as not. That is why I say that you are making unwarranted assumptions. Remember, a choice of two outcomes is not necessarily 50/50.

I suppose you could argue that the average person doesn't make such an offer and that a person who did make such an offer would likely have motivations different from the average person. Perhaps but again you would be making assumptions not presented in the original hypothetical. Which by the way is just a hypothetical that doesn't call for additional premises or contemplation of such premises. If you wish to refine the hypothetical that is fine but as it is it doesn't call for such assumptions. IMO.

I'd switch.
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Anecdotal evidence to be sure, but in my experience, without explicitly stating the classical assumptions, most people I've asked this question assume that Monty would only offer the switch if they picked the car right the first time. If it is possible, the probability of it being true reduces the probability of the car being behind the unchosen door to less than 2/3 and the puzzle becomes an argument about how likely it would be for Monty to have that motivation.
 
Paul C. Anagnostopoulos said:
Folks, the correctly stated "Monty Hall Problem" makes it clear that Monty knows what's behind the doors and shows you one with a goat:
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Yes, but it leaves open the question of why he offered the switch. All we know is that in this one trial, he offered it, but we don't know whether he always does this, or he's trying to trick us / make the show more exciting.

I just now see that Wikipedia has a page that explains it all.
 
CurtC said:
Yes, but it leaves open the question of why he offered the switch. All we know is that in this one trial, he offered it, but we don't know whether he always does this, or he's trying to trick us / make the show more exciting.
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It does make it clear that he won't offer you the other door because you picked the car, which is the important part.
 
tsg said:
Anecdotal evidence to be sure, but in my experience, without explicitly stating the classical assumptions, most people I've asked this question assume that Monty would only offer the switch if they picked the car right the first time. If it is possible, the probability of it being true reduces the probability of the car being behind the unchosen door to less than 2/3 and the puzzle becomes an argument about how likely it would be for Monty to have that motivation.
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Agreed. I think that the puzzle is in the telling. When I tell it Monty isn't the host and no one is playing the price is right. I do reference the show for a visual image but move on and I become the host. Perhaps it is this variable that is significant.

I'm happy to concede that the puzzle can be problematic. However the problems are but semantics and ambiguities in the puzzle as most agree. Sadly these side issues detract from what is otherwise a great puzzle. The point is that puzzle is just a hypothetical. Why should anyone entertain the notion that puzzle can only be considered by the popular version? There are no rules for how one presents the puzzle. Eliminate the superfluous and leave only the relevant parts.

Perhaps it is a matter of perception, I see this "what is the host thinking" issue as so beside the point.

RandFan
 
Number Six said:
The version Paul A used is:

A thoroughly honest game-show host has placed a car behind one of three doors. There is a goat behind each of the other doors. You have no prior knowledge that allows you to distinguish among the doors. "First you point toward a door," he says. "Then I'll open one of the other doors to reveal a goat. After I've shown you the goat, you make your final choice whether to stick with your initial choice of doors, or to switch to the remaining door. You win whatever is behind the door."

This version makes it clear that the host is telling you _beforehand_ that he is going ask you if you want to switch. That makes all the difference. Most or all versions of this I've seen didn't do that. Also, this version doesn't say that you know that there is a car behind one door and a goat behind each of the other two, although just about every other version I've seen does.
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Gotcha, ok, cool.
 
The easiest way to understand this I've found (or thought up, actually) was to imagine not 3 doors but 100.

Clearly if you pick one, the chances are not good you've got the one and only prize. Now Monte can open another door -- remember, he's not opening one at random, but he knows where the prize is. So he avoids it (if it's in the other 99) or doesn't care (if you happen to have nailed it.)

Ok, now he's shown you one. There are still 98. Do you stick with your one, or go with the 98?

The 3 doors problem is akin to you deciding to switch not to "the other one door", but "the other 98". By switching, you inherit not just 98/100, but a full 99/100 chance, since Monte always shows you a non-prize door, since he knows which one it is.

Similarly, with just 3 doors, you have a 1/3 chance, and the other two (cumulatively) have a 2/3 chance. Now he shows you one door, which, guaranteed, will not have the prize behind it. The 2/3 chance thus does not disappear from those two doors because you've gained no information. However, you have gained the ability to inherit the full 2/3 chance with just one door, since the other one's contents are known to you.
 
tsg said:
Interesting puzzle. Marylin is right. Roger is commiting the same error as the Monty Hall problem by assuming that two heads and one head-one tail are equally likely.
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I don't get this coin puzzle. I can see why it is better to switch in the Monty Hall problem, especially with the example of the 100 doors, but not in the one with the coins.

If I had asked person A whether the first coin was a head and they had answered yes, then I would say sure, the chance of the other one being a head is 50%. But since they just volunteered the information, and we don't know why, then it dosn't really tell us anything. In fact most people would only say that the first coin was a head if the other one wasn't, so I would tend to assume that person A's other coin was a tail.

As for person B, I am happy that the chance their other coin is a head is 33%.

Guess this means I don't agree with Roger or Marilyn. :)

But am I missing something here?
 
Wobble said:
If I had asked person A whether the first coin was a head and they had answered yes, then I would say sure, the chance of the other one being a head is 50%. But since they just volunteered the information, and we don't know why, then it dosn't really tell us anything. In fact most people would only say that the first coin was a head if the other one wasn't, so I would tend to assume that person A's other coin was a tail.
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Taken strictly, as is pretty standard in these kinds of logic problems, "the first coin I tossed was a head" only tells us about the first coin and nothing about the second. But, yes, I'm inclined to agree that the tosser wouldn't bother specifying the first coin was a head if they both were, especially since the question was phrased "is at least one of the coins heads-up". But then that makes it more of a language puzzle than a probability one, which isn't really the point. I suppose it could be worded better by asking A "was your first coin heads-up" and asking B "was either coin heads-up".
 
Wobble said:
I don't get this coin puzzle. I can see why it is better to switch in the Monty Hall problem, especially with the example of the 100 doors, but not in the one with the coins.

If I had asked person A whether the first coin was a head and they had answered yes, then I would say sure, the chance of the other one being a head is 50%. But since they just volunteered the information, and we don't know why, then it dosn't really tell us anything. In fact most people would only say that the first coin was a head if the other one wasn't, so I would tend to assume that person A's other coin was a tail.

As for person B, I am happy that the chance their other coin is a head is 33%.

Guess this means I don't agree with Roger or Marilyn. :)

But am I missing something here?
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you're not really missing anything, you're just applying psychology to an information problem. Don't think about it as that person volunteering that information. Instead think about it as this being the information you have. Think about the mathematical probability, not the motives.
 
tsg said:
Taken strictly, as is pretty standard in these kinds of logic problems, "the first coin I tossed was a head" only tells us about the first coin and nothing about the second. But, yes, I'm inclined to agree that the tosser wouldn't bother specifying the first coin was a head if they both were, especially since the question was phrased "is at least one of the coins heads-up". But then that makes it more of a language puzzle than a probability one, which isn't really the point. I suppose it could be worded better by asking A "was your first coin heads-up" and asking B "was either coin heads-up".
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Except that if you word it that way it is obvious that you are being given different information by the two people, and I doubt that anyone who knew much about probability would argue with the answer.
 
Lamuella said:
you're not really missing anything, you're just applying psychology to an information problem. Don't think about it as that person volunteering that information. Instead think about it as this being the information you have. Think about the mathematical probability, not the motives.
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But I don't know how to think about probability for something that only happens once. And if it is happening more than once, then you have to know under what circumstances you will told the first coin was a head. Otherwise how can you decide anything at all? If it was say a computer simulation and you didn't know what rule it was using, then I would say the chance of the other coin being a head was 1/3, just like for person B.
 
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