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Comment by jhncls | Hacker News Reader

Something that really adds to the confusion, is that "in a not mathematically perfect" world, the observer doesn't know whether the host will always open a door. When the candidate chooses the winning door, the host could always open another door, and when the candidate chooses a loosing door, only open another door with a reduced probability. Life teaches people to always be suspicious when free money is promised.

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The "Monty from Hell" scenario (Monty only offers the choice if you picked the car, and otherwise does nothing), which the original statement of the problem allows, is IMO the most likely in real life because it would make for the most entertaining television.

You have real world knowledge of what makes entertaining TV, and perhaps how this particular show actually operated. And you are correct. In the real world, Monty Hall always revealed the worst of the two prizes that the contestant did not select. In the real world, you can use common sense and assume that Monty Hall will use that algorithm.

But in a mathematical probability problem, you can't do that. For example, if there's a slightly smaller object and a slightly larger object concealed in two different sized boxes, common sense says that a person would put the larger object in the larger box. But if it's a abstract mathematical probability problem, and you have no information on the algorithm used, then you have to assume the concealer flipped a coin when they decided which box to use.

If I'm the one setting up the big-small game, and I use the random algorithm, the smaller object is going to be in the bigger box about half the time. If my mom is the one setting up the game, the smaller object is going to be in the smaller box every time.

In abstract mathematical probability problems, if you don't know what algorithm was used, you aren't allowed to bring human behavior in, unless it's stated in the problem.

The problem depends on human behavior, so as originally stated it is unsolvable. You need to clarify that Monty always opens a door, and always reveals a goat.

> But in a mathematical probability problem, you can't do that. For example, if there's a slightly smaller object and a slightly larger object concealed in two different sized boxes, common sense says that a person would put the larger object in the larger box. But if it's a abstract mathematical probability problem, and you have no information on the algorithm used, then you have to assume the concealer flipped a coin when they decided which box to use.

I think that this misrepresents mathematical reasoning.

For solving a problem in a textbook (which a mathematician does intensively at the beginning of his or her career, and seldom thereafter), you must assume neither—that the larger object goes in the larger box, nor that the concealer flipped a coin; you must use only the information contained in the problem. (From this point of view, many problems in most probability textbooks are ill posed, because they force you to make some assumption while solving them; and it then becomes a mind-reading exercise of whether you can successfully make the same assumptions as the poser, rather than a mathematical exercise.)

For a mathematician who actually wishes to use theoretical probability to do something useful, then you must again make assumptions that allow you to translate the messy real world into an idealised mathematical object; but to say that you must, or must not, assume any particular thing is otiose—different assumptions will lead to different results, and the only guide to whether you made the 'right' assumptions is whether or not the idealised mathematical results you get match the real world (to within whatever bounds of error are tolerable).

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