The Monty Hall Problem

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Callum Todd
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The Monty Hall Problem

Post by Callum Todd »

I came across this interesting logic puzzle/problem/paradox thing in a book the other day and while I have actually read about it before I had forgotten it in the ten years or so since then so it impressed me all over again. I searched to see if it had been discussed on this forum before and indeed it has, but also not for ten years or so and not in a dedicated thread so I think this thread is warranted.

For the full details of it see here but I think it would be more fun if people took part on this thread first before reading up the solution and explanation.

I'd also be interested to see how important you think the wording of the problem is. I was very stumped by it at first but on reflection I think the version I read of it missed a crucial detail but perhaps I'm just still failing to fully understand the logic of it. So here goes the puzzle:

---

You are on a gameshow and your objective is to win a prize. On this occasion, the prize is a brand new sports car! The host, Monty Hall, shows you three doors. The doors are numbered 1, 2, and 3. Behind one of these doors is the prize car. Behind the others there is nothing, or just some rubbishy token prize. You have to choose a door, the idea being that you win whatever is behind that door. You choose door 1. Monty then opens door 2, revealing that there is nothing behind it. He now offers you a choice: stick with your choice of door 1, or switch to door 3. This decision will be final. Once you make it, Monty will open the door you have chosen and you will win whatever is behind it.

So do you stick with door 1, or switch to door 3?

---

Would be interested to hear your answers and reasoning behind it. For those that are already familiar with this problem, would be interesting to discuss it once a few answers have been posted.
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Gavin Chipper
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Re: The Monty Hall Problem

Post by Gavin Chipper »

Wow. A fresh Monty Hall problem thread in 2022.
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Callum Todd
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Re: The Monty Hall Problem

Post by Callum Todd »

Gavin Chipper wrote: Thu Feb 17, 2022 6:25 pm Wow. A fresh Monty Hall problem thread in 2022.
Yeah I'm a bit late to the party but then again this is a phpBB forum about an old TV word game show so I'll forgive myself the unhipness.
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Re: The Monty Hall Problem

Post by Fred Mumford »

Callum Todd wrote: Thu Feb 17, 2022 5:46 pm I'd also be interested to see how important you think the wording of the problem is
It's critical. The way you have worded the puzzle, switching would not necessarily change your chances. You have to emphasise that Monty will always open an empty door.

For me the biggest puzzle about the Monty Hall problem is why so many otherwise very intelligent (ie way beyond my level) people seem to struggle with grasping it.
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Re: The Monty Hall Problem

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Fred Mumford wrote: Thu Feb 17, 2022 8:31 pm
Callum Todd wrote: Thu Feb 17, 2022 5:46 pm I'd also be interested to see how important you think the wording of the problem is
It's critical. The way you have worded the puzzle, switching would not necessarily change your chances. You have to emphasise that Monty will always open an empty door.
Yes, that's what got me. I worded it that way because that's how it has been worded in all the versions I have seen. When I first read it I refused to accept the answer and I'm still not convinced that the answer should be accepted if it isn't stated that Monty isn't opening any door at random. But most versions I see don't state that, so is it really necessary? Are we supposed to assume Monty knows where the prize is and won't reveal it? Is that part of the puzzle, or is it a bug?
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Re: The Monty Hall Problem

Post by Jon O'Neill »

Any representation of paradox that doesn't state that he reveals what he knows to be an empty door/goat is simply wrong.
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Re: The Monty Hall Problem

Post by Callum Todd »

Jon O'Neill wrote: Fri Feb 18, 2022 7:14 am Any representation of paradox that doesn't state that he reveals what he knows to be an empty door/goat is simply wrong.
I thought that but then I couldn't understand why most representations I could find didn't state that. It seemed that maybe part of the puzzle was to infer that from the context of Monty being the game show host. Kind of spoils it for me if that inference isn't part of the puzzle, as otherwise it's either wrong (as you say) or too easy and not really that impressive.
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Re: The Monty Hall Problem

Post by Fred Mumford »

There's similar issues with that question about someone who has 2 kids and at least 1 is a boy - what is the probability of them both being boys? Clearly the answer isn't particularly interesting if it's 1/2, so they want the correct answer to be 1/3 - but that entirely depends on how the question is worded and how you know that 1 is a boy.
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Re: The Monty Hall Problem

Post by Ian Volante »

For interest: viewtopic.php?t=47
meles meles meles meles meles meles meles meles meles meles meles meles meles meles meles meles
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Re: The Monty Hall Problem

Post by Callum Todd »

Ian Volante wrote: Fri Feb 18, 2022 1:11 pm For interest: viewtopic.php?t=47
I see Jon phrased it in a way that made clear Monty knows where the prize is. Best part of that thread is Dinos talking about the name Monty in 2008 :)
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Re: The Monty Hall Problem

Post by Noel Mc »

Wanted somewhere to share a revelation I had when I was telling a group of pupils about the Monty Hall problem.

Let me change the scenario slightly:
Three doors, one door has a load of money behind it and the other two are empty.

You pick a door. The host then says:
'Keep that one door, and win its contents, or else you have have both of the other doors, and win the contents behind them.

In this case, it's obviously a no brainer to pick two doors over your one door.

If you then embellish everything, including the host showing an empty door, that's all irrelevant. It's still essentially: do you want one door or two?

Looking for some confirmation from some sage people that this explanation is valid. I think it is sound, but let me know what you think!
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Re: The Monty Hall Problem

Post by Gavin Chipper »

I think it is sound. It's just a case of convincing people that the embellishment of opening an empty door is equivalent to this.

I quite like extending to 100 doors and seeing if people still stick after you open 98.
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Re: The Monty Hall Problem

Post by Noel Mc »

Gavin Chipper wrote: Wed May 15, 2024 1:17 pm I quite like extending to 100 doors and seeing if people still stick after you open 98.
I like this extension, definitely gonna use it!
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Re: The Monty Hall Problem

Post by Tal Lessner »

I know the mathematic rationale behind it and 100% agree with it.
But there's a problem with the wording of the question.
If it was worded like that "You are told to choose a door, and informed that after choosing, the host will reveal an empty door" then no doubt, switch doors, no problem.
But in this case (and every other time I heard / read this problem) it's worded that only after choosing a door, the host decides to be generous and reveal an empty door, at which case, we are no longer in the purely mathematical realm, but also required to understand the host's motives. Maybe the host does the empty door reveal only when the contestant got it right the first time in order to trick his math savvy contestant?
This problem is a cool mathematical exercise, but it's always worded in a non mathematical way which ruins it.
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Re: The Monty Hall Problem

Post by Gavin Chipper »

The first time this problem was presented to me was with no reference to any context about the guy knowing what's behind the doors. So I said it was 50/50, which was apparently "wrong" but with no explanation. But really I wasn't wrong.

But in this thread, it was clarified quite early on by Fred Mumford.
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Re: The Monty Hall Problem

Post by Tal Lessner »

Gavin Chipper wrote: Mon May 20, 2024 5:48 pm But in this thread, it was clarified quite early on by Fred Mumford.
Fred stated it, he also mentioned that on the original problem as presented by Callum, as well as every time I heard it over the past 25 years or so, it is not clear that the host will show an empty door regardless of what you pick.
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Re: The Monty Hall Problem

Post by Gavin Chipper »

Tal Lessner wrote: Tue May 21, 2024 6:40 am
Gavin Chipper wrote: Mon May 20, 2024 5:48 pm But in this thread, it was clarified quite early on by Fred Mumford.
Fred stated it, he also mentioned that on the original problem as presented by Callum, as well as every time I heard it over the past 25 years or so, it is not clear that the host will show an empty door regardless of what you pick.
Yeah, Fred clarified after Callum asked about the importance of the wording of it. It wasn't presented as being the precise canonical version. I don't know why you're so bitter about it.
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Re: The Monty Hall Problem

Post by Tal Lessner »

Gavin Chipper wrote: Wed May 22, 2024 10:20 am I don't know why you're so bitter about it.
Why bitter sweetheart?
Just mentioned that every time I was presented with this problem (many many times) it's always worded incorrectly and agreed with Fred.
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Re: The Monty Hall Problem

Post by Gavin Chipper »

Tal Lessner wrote: Wed May 22, 2024 11:54 am
Gavin Chipper wrote: Wed May 22, 2024 10:20 am I don't know why you're so bitter about it.
Why bitter sweetheart?
Just mentioned that every time I was presented with this problem (many many times) it's always worded incorrectly and agreed with Fred.
You just seemed a bit angry at how Callum had worded the initial post when it had been resolved.
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Re: The Monty Hall Problem

Post by Tal Lessner »

Gavin Chipper wrote: Wed May 22, 2024 12:32 pm You just seemed a bit angry at how Callum had worded the initial post when it had been resolved.
No, just stated it was wrong, why would I be angry about it and what did I say that seemed that way? And I wouldn't have mentioned it had this discussion not been revived last week. But even in the latest version from Noel:
You pick a door. The host then says:
'Keep that one door, and win its contents, or else you have have both of the other doors, and win the contents behind them.
Again, the wording goes back to it not being clear that the host will act that way regardless of your choice, but only after you choose one. That's why I mentioned the wording again.

And this seems like an important bit. Because it's a brilliant logical problem, ruined (every time I encountered it, including when it was worded fully here) by the way it's presented.
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Re: The Monty Hall Problem

Post by Gavin Chipper »

Tal Lessner wrote: Wed May 22, 2024 12:48 pm
Gavin Chipper wrote: Wed May 22, 2024 12:32 pm You just seemed a bit angry at how Callum had worded the initial post when it had been resolved.
No, just stated it was wrong, why would I be angry about it and what did I say that seemed that way? And I wouldn't have mentioned it had this discussion not been revived last week.
I suppose it was because you felt the need to state that it was wrong when it had already been covered. And when I said that Fred had clarified it, you seemed to make a point of saying that Fred "stated" it and that he mentioned that Callum's presentation of it was unclear as if to say "No, Fred did not merely clarify it. He restated it and quite rightly pointed out that Callum was wrong." I mean, that was how it all read to me anyway. But, you know, sometimes we're all guilty of making hasty assumptions about other people's posts and the motivations and thought processes behind them.
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Re: The Monty Hall Problem

Post by Tal Lessner »

Gavin Chipper wrote: Wed May 22, 2024 2:45 pm I suppose it was because you felt the need to state that it was wrong when it had already been covered. And when I said that Fred had clarified it, you seemed to make a point of saying that Fred "stated" it and that he mentioned that Callum's presentation of it was unclear as if to say "No, Fred did not merely clarify it. He restated it and quite rightly pointed out that Callum was wrong." I mean, that was how it all read to me anyway. But, you know, sometimes we're all guilty of making hasty assumptions about other people's posts and the motivations and thought processes behind them.
Fair enough.
I agree 100% with Fred,
I think the wording of Callum & later Noel are a big issue which for me, at least, ruin this problem.
And I felt like stating that the wording of this problem is almost always problematic to say the least.

No feelings (anger, bitterness or otherwise) involved, I just enjoy math problems & a good argument (Which I think you do too).
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Re: The Monty Hall Problem

Post by Gavin Chipper »

Cool.
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