Monty Hall/Take A Deal

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Jon Corby
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Monty Hall/Take A Deal

Post by Jon Corby »

"Take a Deal" (also known as "Monty Hall") was mentioned in the Mathmo test.

The traditional game goes thus: There are three doors: A, B & C. Behind only one of these doors is the star prize - the car, behind the other two doors - nothing. You select a door (let's say A), and then the host, Monty (who knows the location of the prize) always opens one of the other two doors (let's say B) to reveal that it is empty. He then offers you the opportunity to either stick with your initial choice (A), or switch to the other remaining unopened door (C). Is it a straight 50/50 shot, or can you employ a strategy to maximise your chance of winning the prize?

Once you've thought about that, consider if this situation is different:
You select your door (let's say door A). After some initial banter ("are you sure?") etc, Monty is about to walk over and do his door-opening bit. Suddenly, one of the doors swings open (let's call it door B). Monty hadn't touched it, nor given any indication he was about to open it, the lock just broke and the door came open. There's nothing behind it. There's stunned silence, then some laughter, and the director yells "cut!". He comes over and says "well that was lucky, it wasn't your door, and it also wasn't the car. It would have been a pain to have had to take you out back while we moved the car around. Okay, we're gonna start filming again with Monty walking over - obviously Monty you'll have to open door B, and we'll carry on from there..."

Is this different? What would you do in this situation?
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Michael Wallace
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Re: Monty Hall/Take A Deal

Post by Michael Wallace »

it's a nice question, although I'll refrain from answering given that a) I'm doing an MSc in statistics; and ii) I've already thought about this problem when someone has raised it with respect to deal or no deal and the 'swap or no swap' scenario

not that I watch deal or no deal, or anything, it just comes on after countdown...
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Re: Monty Hall/Take A Deal

Post by Conor »

Is the door that swings open the door Monty was intending to open?
Assuming you don't know, I'm going to say it evens up the odds. (1/2 each)
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Re: Monty Hall/Take A Deal

Post by Dinos Sfyris »

Yep defo sounds American. I assume Monty (jeepers what a Yankee name!) usually opens an empty door for dramatic effect and continuance of the game. Therefore if you pick door A and he knows door C has the car behind it he will probably open door B (Empty GASP!). I can't be bothered to discuss the mathematical process that went on in my head but due to human influence alone door C probably has greater chance of having the car behind it so on this alone I would switch doors.

As for one of the doors opening by accident if you don't know which door Monty was going to open then its just 50/50 chance. btw I enjoy topics like this. Its good practice for my PAS101 on Thursday :)
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JimBentley
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Re: Monty Hall/Take A Deal

Post by JimBentley »

I think I've got this. In the original scenario, you should always switch as your odds of getting the car will increase from 1/3 to 2/3, but in the revised scenario it doesn't matter, it's a straight 1/2 vs 1/2.

In the original scenario, let's say the car's behind door A:

You choose A, Monty opens B or C (doesn't matter which) - you switch, you lose, you stick, you win.
You choose B, Monty has to open C, switch you win, stick you lose.
You choose C, Monty has to open B, switch you win, stick you lose.

So, in two from three cases, you switch, you win.

But with the enforced door-opening, a possibility is removed and there's only two cases left. In one you switch and lose, in the other you switch and win.

Or am I talking complete bollocks?
Conor
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Re: Monty Hall/Take A Deal

Post by Conor »

I agree with you Jim.

For the original problem many people think it's 1/2. One way of illustrating that it's better to switch is this: If there were 1,000,000 doors and the host opened all but 2, the one you selected and another one, then it's clearly better to switch. Same principle, on a smaller scale.
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Jon Corby
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Re: Monty Hall/Take A Deal

Post by Jon Corby »

Rats, you're all too clever.
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Re: Monty Hall/Take A Deal

Post by Gavin Chipper »

Yes, people seem to agree with what I would say. Swap in the normal scenario and it's 50/50 in the other.
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Re: Monty Hall/Take A Deal

Post by M. George Quinn »

I love the Monty Hall problem. When the puzzle was originally posed there where several eminent mathematicians arguing it was 50/50. It's really quite simple to grasp when explained clearly (such as Mr Bentley did) but I think it throws most at first.

George
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Jon Corby
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Re: Monty Hall/Take A Deal

Post by Jon Corby »

M. George Quinn wrote:I love the Monty Hall problem.
Me too :D My dad & I spent somewhere in the region of three hours arguing about it with my brother & brother-in-law at my nan's funeral. It's what she would have wanted.
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Re: Monty Hall/Take A Deal

Post by Mark James »

Just thought I'd mention that the Monty Hall problem was featured on the new series of Mythbusters last night. Not only were they testing it to see if switching does increase your chances, but they were also testing if its true that most people don't switch. They played out the scenario with twenty people and every single one of them stuck with their first choice. Can't believe none of them switched. Even if they had never heard of the thing before.
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