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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)

#211

Post by andrewla »

ghaff wrote: post #106
topaz0 wrote: post #95

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…

There are probably a lot of people at this point who have never watched one of these types of game shows. Maybe if they thought about it, they'd realize that MH probably will always show a goat but it wasn't explicit in the initial question that Marilyn answered. Certainly, that constraint is important to the final solution.

No, he wouldn't always open a door -- sometimes he'd offer a trade; like "you can take $1,000 cash from me now, or have your door".

I wonder if anyone has gone through every episode of "Let's Make a Deal" where this scenario occurs and look at the empirical distribution of switch/not-switch.

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

#212

Post by pxx »

empyrrhicist wrote: post #154
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…

Your re-framing is actually not quite right either (or at least, not complete). Key point: if the host picks the car, the game essentially re-sets. Imagine the extreme scenario with 999 goats and 1 car. You select the first door, the host opens 998 doors at random, leaving your selected door and one other. Two scenarios are now possible: 1. There was a car in the 998 doors that got opened. Tough luck, you lose the ga…

Nope. Write the simulation for 2.

Or look at this case analysis for the three element case. https://news.ycombinator.com/item?id=24714612

You can get intuition by understanding that the cases where the game is ruined are only the cases where you didn't choose the car.

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

#213

Post by svat »

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)

The Monty Hall problem is a thought experiment (based on a real game show). As evidence, see this letter from Monty Hall himself ("no trading boxes after the selection"):

Dear Steve:

Thank you for sending me the problem from "The American Statistician."

Although I am not a student of a statistics problems, I do know that these figures can always be used to one's advantage, if I wished to manipulate same. The big hole in your argument of problems is that once the first box is seen to be empty, the contestant cannot exchange his box. So the problems still remain the same, don't they. . . one out of three. Oh, and incidentally, after one is seen to be empty, his chances are no longer 50/50 but remain what they were in the first place, one out of three. It just seems to the contestant that one box having been eliminated, he stands a better chance. Not so. It was always two to one against him. And if you ever get on my show, the rules hold fast for you -- no trading boxes after the selection.

Next time let's play on my home grounds. I graduated in chemistry and zoology. You want to know your chances of surviving with our polluted air and water?

Sincerely,

Monty

https://web.archive.org/web/20100408200824/http://www.letsma...

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

#214

Post by topaz0 »

empyrrhicist wrote: post #154
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…

Your re-framing is actually not quite right either (or at least, not complete). Key point: if the host picks the car, the game essentially re-sets. Imagine the extreme scenario with 999 goats and 1 car. You select the first door, the host opens 998 doors at random, leaving your selected door and one other. Two scenarios are now possible: 1. There was a car in the 998 doors that got opened. Tough luck, you lose the ga…

As I explained in my reply to your comment on mine, this analysis doesn't work out, even in this very-lopsided case. If the game resets when a car is revealed, then games where you first choice happened to be a car will always complete, and games where your first choice was a goat will almost always reset. On average, of 1000 games, your first pick will be a goat 999 times, and of those, 998 will be reset because MH opens the car and 1 will complete and give you the car if you switch, whereas in 1 in 1000 you will pick the car first, in which case the game is certain to complete but you will lose the car if you switch. Therefore, most games will reset, but of the ones that complete, there is no advantage to switching.

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

#215

Post by jncfhnb »

tshaddox wrote: post #203
jncfhnb wrote: post #115

Earlier quoted context omitted.

Doesn’t matter. If you don’t have a guarantee that he will always open a goat door, his opening of the goat door doesn’t give you information.

The statement of the problem is the guarantee, just like the statement of the problem is the guarantee that there is always 1 car and 2 goats, that you always get to initially choose 1 door, etc.

Nope!

It states you opened a door and then Monty opened a door.

It does not state that Monty would have opened a door if you had picked a different one.

If this is not communicated you don’t gain any knowledge from his reveal.

The typical proper Monty hall formula states this assumption clearly that he will always open a goat door. The original one does not state this.

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

#216

Post by flomo »

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…

In reality, Monty Hall often did not behave in the way the problem describes, that might be the source of confusion surrounding this. (e.g. sometimes he would reveal a prize and imply there might be a better prize behind one of the doors. sometimes there were only prizes and no goats, etc.)

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

#217

Post by c2occnw »

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 always avoid the one with the prize" [0]. So readers were arguing her answer was wrong even given this understanding of how the show works.

[0] https://www.newspapers.com/article/the-missoulian-the-monty-...

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

#218

Post by graeme »

empyrrhicist wrote: post #157
topaz0 wrote: post #95

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…

nvm

Monty never shows the car. That was stated in the original reply from Marilyn, and it should be obvious to anyone who's watched a game show. Why on earth would monty open a door, show you the prize (which you now can't get) and then ask you if you want to switch your door?

No one would design a game show like that, so it means the question only has sense if interpreted as "Monty always opens a goat door".

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

#220

Post by calderknight »

kragen wrote: post #210
calderknight wrote: post #205

Earlier quoted context omitted.

I don't think your interpretation of the sentence is sensible. The sentence mentions that the host knows what's behind the doors. So, if he is allowed to open the door with the car, the problem would become insoluble and would just be about speculating on the host's personality. And it definitely doesn't support the conclusion that the probabilities become 50/50.

it's pretty counterintuitive that his personality enters into it, isn't it?

If the format of the game allows him to show the car to the player, how could his personality not enter into it? In every case where the player picks a goat-door, the host will be presented the option to either reveal the car or the goat. I mean, one can imagine various complicated scenarios in which the host might reveal the car exactly 50% of the time in such cases, but none seem like they can be reasonably arrived at.
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