Marilyn vos Savant and the Monty Hall Problem (2015)

I think the important part of the problem is the unstated assumptions around it, and I suspect some people may see different assumptions than others.

If Monty always opens a door, and uses his knowledge of which door has the prize to ensure that the door he opens is always empty, then you should switch, because he's providing you with extra information about which door has the prize.

However, if Monty changes it up and sometimes doesn't open a door, or opens the door with the prize, the whole problem changes. If he only opens a door and offers you the chance to switch when you've picked the door with the prize, you should never switch of course. If he opens a random door and might reveal the prize (after which he obviously won't let you switch anymore), switching doesn't change your odds. I think a lot of people see it as one of those two situations.

There's the Monty Hall Problem and then there's the Second Order Monty Hall Problem Problem.

That is, did the wording as it was originally printed unambiguously define the problem to be solved, or was it ambiguous enough that some people who got it wrong got it wrong for the right reason?

Like many others, I completely missed that when Monty opens a door to show you a goat, he always shows you a goat. I think under close inspection, the problem doesn't actually allow you to interpret otherwise. But, should a problem like this depend on parsing and identifying the ambiguity, or should the problem be defined with explicit details about the door opening?

Given how many otherwise extremely smart people have misunderstood the problem, I am still unsure whether this was an intentional choice in framing the question. I got it wrong and thought Vos Savant was wrong until I read one of the replies in her follow-up article (yes I read this when it first came out!) said they wrote a computer program and reproduced her results. The program implemention worked because they correctly modelled Monty's behavior. I think when people end up writing the program out the ambiguity becomes more obvious.

The explanation in the article is fine as far as it goes, but I think it's much more helpful to emphasize one key fact, which is that MH has to show a goat, and can't open your first choice door, even if it's a goat. That means that if you did pick a goat the first time around, he can only have revealed the other goat, in which case switching will necessarily give you the car. That happens with probability 2/3.

This is implicit in the "100 doors, MH opens 98 of them" construction discussed in the article but making it explicit makes the idea quite clear and intuitive, even with just 3 doors.

I'm moderately sympathetic to the idea that this constraint is not immediately obvious from how the scenario is described, and could technically be ambiguous in some versions. I do think most people will agree that it makes sense to infer this constraint once you think about it. Obviously if he opened the door that you initially picked and it was a goat, you would know to switch to one of the other two (and have 1/2 chance of finding the car). (Or, more absurdly, he could just open the door with the car and you could choose it directly). Or you could imagine a game show where MH can choose to open a door or not, in which case the problem is not well-posed, because you would need to know how he makes the choice.

I remember reading Marilyn vos Savant every week as a teenager. At the time I half suspected that she wasn't real--just a character made by a team or something. [This was before the internet, so I couldn't just ask Google.]

I'm happy that she's real and as smart as portrayed. But what really impressed me is how calmly she received all the negative feedback and personal insults. No matter what, she never lashed back. I have a feeling that's authentic too.