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Comment by FabHK | Hacker News Reader

Yes, indeed, thanks for digging it up.

You see that Marilyn's answer contains the clarified correct specification ("the host, who knows what’s behind the doors and will always avoid the one with the prize..."), but the question does not fully specify it.

Thus, ignoring the clarification in the answer (which furthermore pertains to a modified problem), one could interpret the question differently: the host randomly opens a door [0], this specific day it happens to contain a goat, should you switch?

And that ambiguity has given rise to so much spilled ink (and, by the way, the misconception that statistic professors don't understand probability theory).

[0] Note that specifying that the host knows what's behind what door doesn't help. They could still pick a door randomly.

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I don't think this complaint stands at all. The original question is stated colloquially, not formally, and vos Savant's interpretation of it is the one that's consistent with the question's language and with how game shows work. Even if other interpretations are possible, hers is the one a normal reader would make, and the one the letter-writer apparently intends.

> And that ambiguity has given rise to so much spilled ink (and, by the way, the misconception that statistic professors don't understand probability theory).

I don't think this stands either - the letters quoted don't say anything about the distinction you're making. Unless they're all fictional or selectively edited, a bunch of PhDs really did get the puzzle wrong.

There's also the cultural context of Monty Hall being a real person who had a real game show, on which he really opened doors with goats behind them. Most readers of her column would have been familiar with the mechanics of the show. And the question doesn't really make sense if there's a chance that he opens the door with a prize, there's no more hidden information in that case.

Did any of those readers of her column, including PhD and mathematicians, that declared her solution wrong do so because they objected to the clarification? (Does it make sense to declare the answer wrong ignoring the answer?)

I think they didn't notice the significance of that clarification (which, again, was given in the context of a modified problem). Reading the exchanges again, though, I see that many respondents were not only mistaken, but also impolite or sexist in accusing Marylin of being wrong.

If you specify the problem carefully, anyone with some training in probability should get the answer right. But clearly back then it was a head scratcher.

I wonder whether there've been other problems like that, or we will encounter similar ones: elementary and easy to understand, yet many people get it confidently wrong, until the correct solution permeates through culture.

The original problem is the one you're thinking of as modified, since the original problem was about a real-world game show, and the rules of the real-world show in question included Monty Hall (the host) opening a door that the player hadn't picked and which did not contain the car. The problem assumed familiarity with the game show, the clarification is for people who didn't watch the show.

The modification I'm referring to is the one Marylin suggests with a million doors, with Monty opening 999,998 of.

> [that clarification] was given in the context of a modified problem

I don’t understand what do you mean by that. The clarification was part of the original answer in the context of the original question. The clarification was stated again and again and again. If that’s a “modified problem” so be it but that was the problem being discussed with her readers.

The modification I'm referring to is the one Marylin suggests with a million doors, with Monty opening 999,998 of.

Ok, so the modification was the "Here’s a good way to visualize what happened." explanation.

I would say that the whole (short) paragraph is intended to clarify why the answer to the question is "Yes; you should switch."

Maybe it was not a good way to visualize what happened after all. Experience shows that some readers were unable to understand the argument.

However, none of them complained about the explanation being wrong because those were two completely different problems (only in the second one the host will always avoid the door with the prize, apparently).

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