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Marilyn vos Savant and the Monty Hall Problem (2015)

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Re: Marilyn vos Savant and the Monty Hall Problem (2015)

#221

Post by jncfhnb »

LegibleCrimson wrote: post #180
bemusedthrow75 wrote: post #118

Earlier quoted context omitted.

> His purposeful behavior could also be random. It doesn't matter. Because the problem explains what happens: he opens a door and reveals a goat. That is crucial information. There's now only one goat and one car left. But the choices have not been shuffled: you're definitely still pointing at your original choice. What were the odds that your original choice is a goat? Two in three. If you picked a goat, what is gua…

It's actually counter-intuitive. I was going to argue on your side, and then I wrote up a quick program that proved me wrong[0]. Let's go through the scenarios, with Goat A, Goat B, and Car C. In the scenario where Monty picks a door purposefully, always selecting a goat, the scenarios are: You picked A, Monty showed you B, and you switch to get C. You picked B, Monty showed you A, and you switch to get C. You picked…

You confused yourself. There are four possible outcomes at the end, but they are not equally likely. So not 50/50.

If Monty always shows a goat, it is undeniably a 2/3 chance to win on switch.

The nuance here being discussed is whether you can assume Monty would have shown you the goat had you chosen a different initial door just because he showed you one this time. If you don’t know that, then you don’t learn anything.

Edit: actually I misread. You just chose to ignore the cases where Monty revealed a car. Which is correct although most people chalk that up as a win or loss.

Re: Marilyn vos Savant and the Monty Hall Problem (2015)

#222

Post by bumbledraven »

University of Toronto stats prof. Jeffrey S. Rosenthal wrote a nice paper called "Monty Hall, Monty Fall, Monty Crawl" (https://web.archive.org/web/20230706235720/https://probabili...) in Math Horizons (Sep 2008). It discusses two variations of the Monty Hall problem that are likely to be answered incorrectly by those who used flawed logic to arrive at the correct answer to the original:

> Monty Fall Problem: In this variant, once you have selected one of the three doors, the host slips on a banana peel and accidentally pushes open another door, which just happens not to contain the car. Now what are the probabilities that you will win the car if you stick with your original selection, versus if you switch to the remaining door?

> Monty Crawl Problem: As in the original problem, once you have selected one of the three doors, the host then reveals one non-selected door which does not contain the car. However, the host is very tired, and crawls from his position (near Door #1) to the door he is to open. In particular, if he has a choice of doors to open (i.e., if your original selection happened to be correct), then he opens the smallest number available door. (For example, if you selected Door #1 and the car was indeed behind Door #1, then the host would always open Door #2, never Door #3.) What are the probabilities that you will win the car if you stick versus if you switch?

Re: Marilyn vos Savant and the Monty Hall Problem (2015)

#223

Post by paulcole »

dekhn wrote: post #189
paulcole wrote: post #55

Earlier quoted context omitted.

I once had somebody claim the answer was wrong because we weren't sure if the contestant wanted the car or the goat. This puzzle is like a shibboleth for nitpickers.

In many developing countries if you ask somebody whether they want a flush toilet or a mobile phone, most people pick the mobile phone.

Do you think that given the context of the puzzle – about a US game show, printed in a US paper that the question is being asked in many developing countries?

Or are you looking to find fault with something for the sake of finding fault?

Re: Marilyn vos Savant and the Monty Hall Problem (2015)

#224

Post by dfxm12 »

apetresc wrote: post #192
bazoom42 wrote: post #186

Earlier quoted context omitted.

> it just wouldn't make sense in the context of the game show for the presenter (who knows where the goats are) to ever open a door to reveal the car and give you the option to switch. None of it makes sense because it is not a real game show, it is a thought experiment.

Apropos the topic of being confidently incorrect, it was indeed a real show: https://en.wikipedia.org/wiki/Let%27s_Make_a_Deal Hosted by a real Monty Hall: https://en.wikipedia.org/wiki/Monty_Hall It even had the gag prizes like goats (or, as in the photo from the wiki page, a llama)

I think importantly, the "Monty Hall Problem" as described is different from how any iteration of the Big Deal at the end of Let's Make a Deal is actually run. Hall himself recognized this and mentioned it in a NYT interview in the 90s. It's paywalled, but the excerpt is here: https://en.wikipedia.org/wiki/Monty_Hall_problem

Hall understood the problem, giving the reporter a demonstration with car keys and explaining how actual game play on Let's Make a Deal differed from the rules of the puzzle.

When I first heard about the puzzle, I struggled with the solution precisely because I kept making assumptions about the puzzle based on my knowledge of the show. It wasn't until I thoroughly read the question, setting aside those assumptions, that I realized it is materially different from the show and I was able to really understand the solution.

Re: Marilyn vos Savant and the Monty Hall Problem (2015)

#225

Post by dekhn »

paulcole wrote: post #223
dekhn wrote: post #189

Earlier quoted context omitted.

In many developing countries if you ask somebody whether they want a flush toilet or a mobile phone, most people pick the mobile phone.

Do you think that given the context of the puzzle – about a US game show, printed in a US paper that the question is being asked in many developing countries? Or are you looking to find fault with something for the sake of finding fault?

No, I'm making the pooint that in many parts of the world, a goat is more valuable than a car because of the lack of car infrastructure.

Re: Marilyn vos Savant and the Monty Hall Problem (2015)

#226

Post by lucas_membrane »

NoboruWataya wrote: post #161

Wow, I knew about the problem and was vaguely aware that it had generated some controversy amongst statisticians but I had no idea about the insanely arrogant and obnoxious (not to mention wrong) abuse Marilyn had received from the "intellectual elite". I have little sympathy for the "ambiguous question" defence. Not only is Marilyn's interpretation grammatically valid, it just wouldn't make sense in the context of t…

>> The game would be reduced to "does Monty feel like gifting you a car or a goat today".

But, in fact, that is the way the real show worked. He was more likely to give you a car when he hadn't given away any cars for a while. And Marilyn's description of the problem included no information about whether or not Monty offered to swap doors rarely, frequently, or always, or about the previously observed outcomes depending on whether or not the contestant took the bait. And she surely did not say that Monty was a robot and had no interest in what happened on his show.

The way the game actually worked is like how poker and life work until proven otherwise. Every high-stakes game has a sucker, and if you do not know who is the sucker, it is you. Read the story in which a NY Times reporter interviewed Monty about the controversy and you will learn how it really worked.

Re: Marilyn vos Savant and the Monty Hall Problem (2015)

#227

Post by flomo »

andrewla wrote: post #206

The reason the scenario is kind of twisted is that in the iterated game of "you pick door, he shows goat, switch or no", vos Savant's answer is clearly correct. But in the iterated game of "you pick door, and host chooses whether to show a goat or to let you open the door you initially chose" I don't think you (or he) can get better than 50-50. If you chose the correct door initially, then he can try to trick you int…

That's correct, the premise of the show was more Monty Hall doing something to challenge the contestant. That's probably why people missed the problem's premise where he always opens a goat door.

Re: Marilyn vos Savant and the Monty Hall Problem (2015)

#228

Post by LegibleCrimson »

jncfhnb wrote: post #221
LegibleCrimson wrote: post #180

Earlier quoted context omitted.

It's actually counter-intuitive. I was going to argue on your side, and then I wrote up a quick program that proved me wrong[0]. Let's go through the scenarios, with Goat A, Goat B, and Car C. In the scenario where Monty picks a door purposefully, always selecting a goat, the scenarios are: You picked A, Monty showed you B, and you switch to get C. You picked B, Monty showed you A, and you switch to get C. You picked…

You confused yourself. There are four possible outcomes at the end, but they are not equally likely. So not 50/50. If Monty always shows a goat, it is undeniably a 2/3 chance to win on switch. The nuance here being discussed is whether you can assume Monty would have shown you the goat had you chosen a different initial door just because he showed you one this time. If you don’t know that, then you don’t learn anythi…

I'm covering the problem statement. I'm ignoring the cases where Monty revealed a car because it didn't happen. You can't chalk it up as a win or a loss, because it didn't happen.

It's always "You choose a door, the host reveals a goat behind another door. Do you switch?" The scenario does not include him revealing the car.

If he intentionally chooses a goat, switching gets the car 2 times out of 3.

If he chose randomly and just happened to choose a goat, switching doesn't matter. It's 50/50.

Of course, these aren't the only scenarios. As others have mentioned, if Monty can decide whether to reveal what's behind a door, he can act maliciously and only reveal a goat if you've already selected a car, and switching will always make you lose. Without knowing these constraints, a single answer isn't knowable, as you mentioned before.

Re: Marilyn vos Savant and the Monty Hall Problem (2015)

#229

Post by mlyle »

c2occnw wrote: post #217
dekhn wrote: post #93

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 c…

That interpretation requires believing that the host, who knows what’s behind the doors, would sometimes open the door revealing the automobile and ask the contestant if they want to switch to that door. It doesn't really make sense, so that should tip off readers that they've misunderstood the show format. Marilyn even made this explicit in her answer, writing that the host "knows what's behind the doors and will al…

> That interpretation requires believing that the host, who knows what’s behind the doors

The host has to always open the door, too. Else-- if your strategy is always to switch when shown a goat, the host can choose to only show you a goat if you have already picked the car (which would cause you to lose 100%). Or various mixed strategies.

This is one of those problems that is hard to grasp at baseline, but a little harder to grasp because the situation isn't fully specified. As a result it tends to create a lot of controversy, like the one about an airplane on a treadmill.

Re: Marilyn vos Savant and the Monty Hall Problem (2015)

#230

Post by habinero »

jncfhnb wrote: post #178
bemusedthrow75 wrote: post #171

Earlier quoted context omitted.

> You do not know if he would have shown door 3 had you picked door 2! This is NOT in the prompt. You cannot assume that. Monty isn't a contestant. Monty's action and outcome is part of the fixed description of the problem. He opens one of the doors and reveals a goat. Full stop. If you think revealing one of the two goats (the other of which might or might not be behind your current selection) isn't information of v…

> Monty isn't a contestant. Monty's action and outcome is part of the fixed description of the problem. He opens one of the doors and reveals a goat. Full stop. Incorrect and I invite you to cite the line of the prompt that says otherwise. This is the critical missing piece of information that makes it the commonly understood “Monty hall problem” and not the improperly stated prompt that we actually have. “Monty did…

No, you're solving the wrong thing. This is a thought experiment, not a real life scenario with ambiguity.

In this thought experiment, MH will always open a door to a goat on purpose and always ask if you want to switch, because the entire point of the thought experiment is that people find the 2/3 odds if you switch unintuitive and baffling, leading to extremely long thread chains.

You're doing the equivalent of saying "why can't we wait and see if the cat meows" when talking about the Schrödinger's thought experiment. It can't because that's not the point of the exercise.

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