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Monty Hall Problem... For Newbies

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  • Total voters
    141
bonzombiekitty said:
The easiest way of explaining the monty hall problem to people who have trouble understanding is to use a million doors instead of 3. I find that people grasp it better when you can say "If you picked the right door at the beginning, you win if you stay with the door you chose. When you chose a door, you had a 1 in a million chance of being right..."

Something about it seems to make it click in people's heads

edit: Fener has already mentioned this.
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That's the part about this that goes over my head and gave me troubles trying to explain it to other people.

Let's say you didn't pick ANY door from the million and he opened all but 2, then you got to choose. That would be 50/50 since you're only choosing between 2 doors. If the premise of the riddle is that he always opens "Goat" doors until you are left with only 2 doors, doesn't it seem inconsequential as to whether or not you had a 1/1,000,000 chance to get it right the first time? By the rules of the riddle you will always have a 50/50 chance because he will always get rid of every wrong option except for one and ... oh crap I think I get it now.

He will always get rid of every wrong option except for one and that means that staying with your door means that you happened to get that 1/1,000,000 chance at the very beginning and he opened all the wrong doors except one and your selected door. It's far more likely that you chose a wrong door and he opened all of the wrong doors except for your wrongly-chosen door and the correct door ... wow that analogy really does help.

So the crux of the riddle is the fact that the host will always open all wrong doors except one, meaning you are always left with a wrong and right door and it's more likely that you chose incorrectly to begin with and he was merely forced make your chosen door the single unopened wrong door.
 
bruto said:
and so, complaining about a hypothetical problem in a hypothetical game because it is "contrived" seems out of place.
If I started a thread asking "what would change if baseball players got four strikes instead of three," that would be the subject of the thread, even though nobody plays that game.
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Isn't that pretty much what I said in the first place? I wasn't complaining about contrivance; I found it amusing that others wanted to talk only the original formulation of the problem rather than any of the variants, when they're all equally contrived.
 
StevenLeonCooper said:
That's the part about this that goes over my head and gave me troubles trying to explain it to other people.

Let's say you didn't pick ANY door from the million and he opened all but 2, then you got to choose. That would be 50/50 since you're only choosing between 2 doors. If the premise of the riddle is that he always opens "Goat" doors until you are left with only 2 doors, doesn't it seem inconsequential as to whether or not you had a 1/1,000,000 chance to get it right the first time? By the rules of the riddle you will always have a 50/50 chance because he will always get rid of every wrong option except for one and ... oh crap I think I get it now.

He will always get rid of every wrong option except for one and that means that staying with your door means that you happened to get that 1/1,000,000 chance at the very beginning and he opened all the wrong doors except one and your selected door. It's far more likely that you chose a wrong door and he opened all of the wrong doors except for your wrongly-chosen door and the correct door ... wow that analogy really does help.

So the crux of the riddle is the fact that the host will always open all wrong doors except one, meaning you are always left with a wrong and right door and it's more likely that you chose incorrectly to begin with and he was merely forced make your chosen door the single unopened wrong door.
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You buy a Lotto ticket with odds of millions to one against winning. You know the Lotto ticket you bought is extremely, highly unlikely to be the winning ticket. If the Lotto people said to you could either keep the (extremely, highly unlikely to be the winning ticket) ticket you bought, or have another ticket that was absolutely guaranteed to be the winning ticket if yours wasn’t, would you switch tickets?

Essentially the Lotto people are saying “Do you want to keep your one ticket or swap it for all the millions of other tickets?”
 
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bobdroege7 said:
Carol Merrill,

I'll admit I had to look it up for the spelling.
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Winner! I was starting to think nobody was going to get it. And I, too, had to check the spelling on the Web.

Monty always called her by her full name. I think got a kick out of the sound of it.
 
I not only got this wrong the first time I heard it (around age 25), but argued pigheadedly for weeks on the particular discussion forum where it was presented.

The solution not only made me question my own presumptions/statistics/"common sense", but also made me question my skepticism in general. Hell, it took the Monty Hall calculator to finally change my mind. Despite actually finally grokking what the stats/host-role were I couldn't "believe" that they were right until even a machine told me I was being pigheaded.

So I then thought "wow, if I'm so pigheaded and willfully wrong about that, what else?" And have hopefully been a much better self-skeptic since. Not to mention other willfully blind areas like politics, ideologies, etc. The Monty Hall problem was a big, big influence on me and my previous narcissistic surety.
 
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marting said:
Good TED discussion.

This fits in
http://imgs.xkcd.com/comics/frequentists_vs_bayesians.png
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While that comic is funny and highlights the importance of priors in certain situations, I have to defend frequentists. Frequentist theoriticians have gone to great lengths to point out the limitations of that approach. And no decent statistician who specializes in frequentist statistics would attempt to apply it to a situation where it was obviously inappropriate. Just as a statistician who specializes in spatial or categorical statistics wouldn't try to apply their techniques to situations in which they made no sense.

Not that people don't misuse statistics all the time, but most of the time those people aren't statisticians.
 
cornsail said:
While that comic is funny and highlights the importance of priors in certain situations, I have to defend frequentists. Frequentist theoriticians have gone to great lengths to point out the limitations of that approach. And no decent statistician who specializes in frequentist statistics would attempt to apply it to a situation where it was obviously inappropriate. Just as a statistician who specializes in spatial or categorical statistics wouldn't try to apply their techniques to situations in which they made no sense.
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Yeah. I could challenge someone to allow me a single 8-spin run of Martingale at roulette and probably win, but that can never justify a claim that the probabilities that undermine Martingale are somehow faulty.
 
cornsail said:
While that comic is funny and highlights the importance of priors in certain situations, I have to defend frequentists. Frequentist theoriticians have gone to great lengths to point out the limitations of that approach. And no decent statistician who specializes in frequentist statistics would attempt to apply it to a situation where it was obviously inappropriate. Just as a statistician who specializes in spatial or categorical statistics wouldn't try to apply their techniques to situations in which they made no sense.

Not that people don't misuse statistics all the time, but most of the time those people aren't statisticians.
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It was the misuse discussed in the TED talk and in exaggerated form by the xkcd cartoon that struck me as similar. In particular the erroneous use of statistics in convicting people of SIDS is disturbing.
 
Dessi said:
My young nephews struggled with the 3-door Monty Hall problem, but it seemed to make a little more sense to them when I explained the scenario with 100 doors instead:

Imagine there are 100 doors, 99 with goats and 1 with a car. You select a door at random; it could be a goat, or it could be a car. Monty Hall then opens 98 doors at random, showing 98 goats. Only two doors remain: do you switch? Hell yes you do.
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I floated that idea 10 years ago. I don't know if I was the first.


The key thing to realize is Monte doesn't pick at random himself -- he knows which of the other two (or both) does not have a car.
 
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marting said:
It was the misuse discussed in the TED talk and in exaggerated form by the xkcd cartoon that struck me as similar. In particular the erroneous use of statistics in convicting people of SIDS is disturbing.
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I completely agree with you on both those points.

My only issue with the comic was the depiction of "frequentist statisticians" as inferior to "Bayesian statisticians". The people misusing statistics are typically not statisticians. And IMO the frequentist and Bayesian approaches both have their place. Whether or not one or the other is appropriate all just depends on the situation.
 
monty_hall.png
 
Beerina said:
I floated that idea 10 years ago. I don't know if I was the first.
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You are not the first. Marilyn vos Savant tried this explanation back in mid-90's.

And some people STILL claimed that with 98 doors open there is no difference between two remaining closed doors!
 
I freely admit that I have not the time to read the entire thread and apologize in advance if this has already been shared.

The way I was able to wrap my head around the Monty Hall problem was to put together a quick and dirty Monte Carlo simulation in Excel, which provides tangible and convincing solution. There are also some online simulators available, as below, which some might find helpful.

http://www.grand-illusions.com/simulator/montysim.htm
 
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