I think it's made pretty explicit. In this version: "Then, the host, who is well-aware of what’s going on behind the scenes, opens door #3, revealing one of the goats." The host has to open a door that doesn't show the car.

Yeah - I don't think that statement clarifies. The rule that the host will never reveal the prize should be made explicit.

You're quoting Priceonomic's 2021 description of the problem. This is NOT the 1990 Parade Magazine description of the problem that generated all the responses.

The 1990 Parade Magazine description is almost identical (and actually more explicit, since "say #3" is a removable parenthetical): "the host, who knows what’s behind the doors, opens another door, say #3, which has a goat" -- https://web.archive.org/web/20130121183432/http://marilynvos...

The host's knowledge is explicitly mentioned, and the only purpose this could have is that so he can use it to avoid giving the game away.

That is literally not explicit (where by "literally" I mean literally, not figuratively, and by "explicit", I mean explicit, not implicit). It is sort of hinted at, but it is not explicitly said that the host will mechanically reveal a door that has a goat. It is quite conceivable that the host picks a door randomly, or in fact that he picks the door with the car with a certain probability (saving the show quite some money).

> It is literally explicit in the 1990 rendition.

> 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

Remove the parenthetical example "say #3", and the parenthetical "who knows what's behind the doors" and that sentence reads: "...and the host opens another door which has a goat".

The "has a goat" is not a hypothetical example. It's a (both literal and explicit) declaration of the rules statement. The rules seem very clear that the host will open a door with a goat.

It matters whether they choose the door because it has a goat vs. they randomly choose a door that happens to have a goat.

It's not explicitly saying the former even when stating the result, because everything is written as a certainty in retrospect. You can say, "the roulette wheel stopped on 00," but it was still a random event when it happened.

nope, it's ambiguous as stated, if you're a literalist.

It's a tv show - a valid purpose is that he needs to stretch by 30 seconds before going to commercial. Another purpose is that the audience thinks seeing a goat is funny. Another purpose is to prove the show uses two different goats and doesn't do a switcheroo behind the scenes.

What the Parade article does NOT say is that the first door opened is ALWAYS not the contestant's choice and ALWAYS reveals a goat.

That he always opens a door that is not the contestant's choice is a necessary implication of saying that he opens "another" door. That he opens a door containing a goat is similarly implicated. It's not even an implication, really; it's just definitional in vernacular English. And in any event, any of the academics writing in would have been perfectly familiar with the semantic structure of such logic puzzles.

Also, most of the population of the country would have been familiar with the game show. Let's Make a Deal was one of the most popular programs on television. Even if they didn't particularly like it, people didn't have much of a TV program selection back then. And even if someone didn't regularly watch TV, it's likely they would have seen it and been familiar w/ the rules regardless simply because it was so widely watched around them. The vehemence of the respondents may even have been an effect of their familiarity, unable to recognize and accept that they'd missed a crucial analytical element after having watched the show so many times before.

What makes the puzzle counterintuitive, IMO, is that you're actually answering two different questions: the first before the host makes his selection, and the second (the only one that matters) after he makes his selection. Developing the habit of rigorously "updating your priors"--that is, recognizing when the available evidence has changed, requiring a reassessment--is an applied skill that even experts aren't particularly good at. (Nor was I the first time I read the problem.) The real-life narrative structure actually makes for a great logic puzzle as people are primed to miss the final constraint (his selection) for what it is. They're used to analyzing static situations, or a series of linked static situations--solve X correctly to solve Y. That makes the Monty Hall Problem peculiarly distinct from the typical type of logic puzzle that simply includes a misleading, irrelevant constraint or requires resolving a complex set of constraints.

> That he always opens a door that is not the contestant's choice is a necessary implication of saying that he opens "another" door

It really isn't. It's possible Monte Hall always opens the doors in reverse numerical order, so if you picked Door #3, that would get opened first. Or that if Door #3 contained the car, that would be revealed before any goats. The way the scenario was initially described by Ms. Vos Savant is not clear about that.

And no, not everybody knew Let's Make a Deal in great detail. It's always been a daytime tv show, meaning, it airs while most people are at work.

> > That he always opens a door that is not the contestant's choice is a necessary implication of saying that he opens "another" door

> It really isn't. It's possible Monte Hall always opens the doors in...

It really is. The "other" in "another" means, quite literally, "not the one already mentioned". And since the only door mentioned before that was the one the contestant had chosen, "he then opens another" not only implies but quite literally and explicitly says that this "another" door the host opens absolutely isn't, cannot be, the one the contestant chose. That's quite simply what "another" literally means.

The problem the reader originally submitted clearly implied both those things, at least clearly enough for Marylin

> and the host, who knows what’s behind the doors, opens another door ... which has a goat

The explanation Marylin originally wrote makes the second assumption explicitly clear to avoid any possibility of confusion

> Suppose there are a million doors, and you pick door #1. Then the host, who knows what’s behind the doors and will always avoid the one with the prize, opens them all except door #777,777. You’d switch to that door pretty fast, wouldn’t you?

People still wrote in claiming she was wrong.

> Since you seem to enjoy coming straight to the point, I’ll do the same. You blew it! Let me explain. If one door is shown to be a loser, that information changes the probability of either remaining choice, neither of which has any reason to be more likely, to 1/2. As a professional mathematician, I’m very concerned with the general public’s lack of mathematical skills. Please help by confessing your error and in the future being more careful.

She then explained the solution a second time with a table clearly illustrating the assumptions and the odds.

People continued to write in claiming she was wrong.

> I’m receiving thousands of letters, nearly all insisting that I’m wrong

> 92% are against my answer, and and of the letters from universities, 65% are against my answer.

Lack of clarity did not generate all the letters.

I run a small website with logic puzzles. I get thousands of emails. Some tell me the problem is misleading. Some tell me the answer is wrong.

The one thing I have learned is that once someone's mind is made up, it will never change.

The other thing I have learned, is that a good problem is succinct.

The smartest people in the room will ask clarifying questions to test out their assumptions before they even attempt an answer.

Yes.

The probability that car is behind the door NOT selected by the host is 1/2 - because the host had only two choices which are equally good for him.

Probability that car is behind the door initially selected by the user is 1/3 because that is what happens when you randomly choose one out of three.

We have to think in terms of two different probabilities: a) That user selects the correct door initially and b) That door not selected by the host has the car.

But yes it is counter-intuitive. I can't quite point out which step in the flawed reasoning of no benefit from switching occurs. How do people come up with the incorrect result, and what are the flawed assumptions they make that cause them to arrive at the wrong answer? Where exactly does the logical error happen?

People come up with the incorrect result because with 2 doors remaining, it would initially appear that they both have equal chance of containing the car. Because it is unclear HOW we arrived at the final two doors. If we arrived there randomly, the probability is in fact 50-50. If there was some manipulation - i.e. if the host knows where the car is AND deliberately opens doors that do NOT contain the car - then the probability is not 50-50, and switching away from the initially-chosen door is advantageous.

I came up with a slightly different reasoning why you should switch: When the host is about to open a door there are 2 possibilities, he can either choose between 2 goats, OR he can choose between a car and a goat.

In more than 50% of the cases the host encounters the situation where he must choose between a car and and goat. Why because in > 50% of the cases the user's initial choice does NOT have car behind it, so the host must choose between a goat and a car in > 50% of the cases.

And of course he chooses the goat. So in > 50% of the cases the remaining door will have the car behind it,

So by switching, you have > 50% chance of winning the car. Is this the correct answer? It would seem to me your chance to win if you switch is 1/2 x 2/3. Is this correct?

What is counter-intuitive is why the action by the host should give us any information about which door has the car. And it does not do so always. It only gives us information in 2/3rds of the cases. If the host chooses between two goats that should not give us any information at all should it? But that is good enough for us since we play the odds, and switching seems to increase our odds.

But, now I have my doubts. This kind of probabilistic reasoning would seem to apply only if we repeat the experiment many times. But no contestant gets to play this game more than once. Probability theory does not tell us anything about what happens in a single experiment. So is it rational for the player to think they should switch if they can play this game only once?

How much should they be willing to pay for the option of being able to switch the door, if they know they can only play this game once?

I find it a little more intuitive in a visual way with the 100 doors. If the 98 doors that the host subsequently eliminates leaves one in the middle of the pack (e.g. #52), most rational people would look at that door and say "hmm its probably in there rather than in the one i picked, #1"

Except how many people, when presented with option of picking one of a hundred doors, picks Door #1?

Further, the problem with the 100-door (or 1000, or million, etc) rationalization is that it did not appear in the initial Parade Magazine article which generated all the negative responses. Sure, today you know the secret and can explain the problem well, but it isn't fair to judge the people whose only exposure to the issue was the first, original Parade Magazine article.

Eh, the formulation I like goes:

You pick a door.

The host runs up and knocks on one of the other doors, without opening it or saying anything. He just indicates a door.

Now you have the option to stick with your one original door, or to take both of the other two doors.

Which is better?

And then: how is that scenario different from the one where he opens the special door and reveals a goat?

I read this and thought that sometimes, the host would open the door with the car.

Marilyn had the same framing of the problem as everyone else.

Marilyn made her key assumptions explicit in the original explanation she gave.

92% of people still disagreed with her.

And most of them seemed fully aware that she had the guiness world record for the highest IQ in the world.

To paraphrase Paul Graham's recent article, If the smartest person in the world proposed an idea that sounded preposterous, I'd be very reluctant to say "You are wrong."