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>, and if in this particular run through he happens to open a door with a goat, then you will still double your odds if you switch. Yes, I agree. >The host d

by jasode 9y ago

>, and if in this particular run through he happens to open a door with a goat, then you will still double your odds if you switch. Yes, I agree. >The host doesn't need to know what's behind the door for this to work! The host has to know it and always open the goat door to prevent mathematicians from making a repeated simulation in their minds where the host sometimes opens a door with a car behind it. That would be over-reading and misinterpreting the MH wording but that's why the problem is so enduring -- it uses clever text to misdirect a lot of smart people away from the correct 2/3rds answer.

deleted 9y ago
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yen223 9y ago
Weirdly enough, in the scenario where the host doesn't know which door has the goat and just happens to open the right door, your odds of winning the car if you switch actually does go back to 50-50. Edit: I made a quick IPython notebook to demonstrate this: https://github.com/yen223/monty_fall/blob/master/Monty%20Hall%20vs%20Monty%20Fall.ipynb https://github.com/yen223/monty_fall/blob/master/Monty%20Hal...
- mannykannot 9y ago
  In your program, you have to classify all cases where the host picks the prize on his second selection as 'passed', and put them aside. This affects the result, as in all those cases the subject had picked one of the wrong boxes, so you have biased the cases you actually consider towards those where the subject originally picked the right box.
Scarblac 9y ago
In the case where he just opens random doors, it doesn't matter that you switch, your door and the other unopened have the same chance. After all, it'd be the same if you chose the door to be opened. If that mattered, you should always mentally choose a door, choose another to randomly open, switch and make that your choice. That's absurd. The whole thing is that be cause the host has to open a door with a goat, switching means you basically get to choose both other doors.
dalbasal 9y ago
I think if the mathematician (or anyone) gets to that point, he's solved the problem even if the answer is in the form of a probing question. There might be confusion about whether or not the host "cheats" and always shows a goat, but if you're asking this question you get to the answer. The issue is that people don't get to this question. I think that's not a wording-misdirection thing, it's something about how people think.
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mannykannot 9y ago
To be clear, as we are all in agreement that the correct answer is the same whether the question states the host has knowledge or not, there is no doubt here that the question, exactly as stated, is unambiguous and has one correct answer. If someone chooses to use faulty reasoning in solving it, they have made a mistake as surely as if they had made an arithmetical error in calculating the odds incorrectly. Mathematicians and philosophers, in particular, are supposed to know how to be rigorous. I think this has wandered across the threshold into "if they were asked a different question, they would have given a different answer", which might be true, and, if tested experimentally, might throw some light on why this problem seems unreasonably baffling, though it would be hard to find sufficient suitable candidates for this experiment (maybe among schoolchildren who have shown they have a good grasp of the relevant statistics). Personally, I would guess that people are led down the wrong path by causal intuition - i.e. "opening the second box makes no difference to where the prize is", or perhaps by an intuition that the choice to switch or not is as random a choice (i.e. over uniform odds) as their initial pick. I realize that at this point, these guesses have no more evidence for them than does the hypothesis that people are being misled by language clues.

by jasode 9y ago

deleted 9y ago

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yen223 9y ago

Weirdly enough, in the scenario where the host doesn't know which door has the goat and just happens to open the right door, your odds of winning the car if you switch actually does go back to 50-50. Edit: I made a quick IPython notebook to demonstrate this: https://github.com/yen223/monty_fall/blob/master/Monty%20Hall%20vs%20Monty%20Fall.ipynb https://github.com/yen223/monty_fall/blob/master/Monty%20Hal...

mannykannot 9y ago
In your program, you have to classify all cases where the host picks the prize on his second selection as 'passed', and put them aside. This affects the result, as in all those cases the subject had picked one of the wrong boxes, so you have biased the cases you actually consider towards those where the subject originally picked the right box.

Scarblac 9y ago

In the case where he just opens random doors, it doesn't matter that you switch, your door and the other unopened have the same chance. After all, it'd be the same if you chose the door to be opened. If that mattered, you should always mentally choose a door, choose another to randomly open, switch and make that your choice. That's absurd. The whole thing is that be cause the host has to open a door with a goat, switching means you basically get to choose both other doors.

dalbasal 9y ago

I think if the mathematician (or anyone) gets to that point, he's solved the problem even if the answer is in the form of a probing question. There might be confusion about whether or not the host "cheats" and always shows a goat, but if you're asking this question you get to the answer. The issue is that people don't get to this question. I think that's not a wording-misdirection thing, it's something about how people think.

deleted 9y ago

[deleted]

mannykannot 9y ago

To be clear, as we are all in agreement that the correct answer is the same whether the question states the host has knowledge or not, there is no doubt here that the question, exactly as stated, is unambiguous and has one correct answer. If someone chooses to use faulty reasoning in solving it, they have made a mistake as surely as if they had made an arithmetical error in calculating the odds incorrectly. Mathematicians and philosophers, in particular, are supposed to know how to be rigorous. I think this has wandered across the threshold into "if they were asked a different question, they would have given a different answer", which might be true, and, if tested experimentally, might throw some light on why this problem seems unreasonably baffling, though it would be hard to find sufficient suitable candidates for this experiment (maybe among schoolchildren who have shown they have a good grasp of the relevant statistics). Personally, I would guess that people are led down the wrong path by causal intuition - i.e. "opening the second box makes no difference to where the prize is", or perhaps by an intuition that the choice to switch or not is as random a choice (i.e. over uniform odds) as their initial pick. I realize that at this point, these guesses have no more evidence for them than does the hypothesis that people are being misled by language clues.