mkl654321
mkl654321
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August 8th, 2010 at 11:27:59 PM permalink
I've just read the old series of posts on the Monty Hall problem. The explanations--here and everywhere I've looked--consist of equations (albeit simple), which have never been satisfying to me, and I suspect, to many other people, as explanations. What seems itchy about the explanations that are offered is that we know that Monty will reveal a goat, and that we also know that there will be at least one goat for him to reveal. So it seems counterintuitive that something he will ALWAYS do, no matter what, will affect the outcome. Here's how I decided to restate the apparent paradox in English, rather than mathese:

You pick, let's say, Door 1. Monty now reveals Door 2 (or 3), behind which is a goat. The question is:

Did Monty reveal Door 2 (or 3) because he HAD to (you picked the other goat), or because he CHOSE to (you picked the car)? Since you would have picked a goat two out of three times, logically, two out of three times, Monty was FORCED to pick the remaining goat-door, and therefore you should switch. In other words, it is twice as likely that Monty chose to open the door that he did because he was FORCED to, rather than having a choice and happening to choose that one.

It seems to me that this is very similar to the problem of Restricted Choice in bridge.
The fact that a believer is happier than a skeptic is no more to the point than the fact that a drunken man is happier than a sober one. The happiness of credulity is a cheap and dangerous quality.---George Bernard Shaw
DJTeddyBear
DJTeddyBear
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August 9th, 2010 at 5:00:32 AM permalink
I've never bought into the 'math', BECAUSE as you point out, Monty doesn't always open a random door. If you switch, how does your 1:3 chance become 2:3? The 2:3 includes and exposed goat!?!? I ALWAYS thought that the option to switch after exposing the goat makes it a 1:2 situation. Until...

It was explained this way: If there were 1,000 doors, and Monty opened 998, you KNOW that the prize is behind either your door or the last door. Do you REALLY think you were lucky enough to have picked correctly when it was a 1:1000 chance? So, even though there are 998 goats exposed, switching makes your chance 999:1000. The same logic applies to the standard three door scenario.

If Monty opened a door at random, then normal 1:2 logic applies. Similarly, in Deal or No Deal, since suitcases are opened at random, normal logic applies.

I don't know Bridge, so I can't comment on that.
I invented a few casino games. Info: http://www.DaveMillerGaming.com/ 覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧 Superstitions are silly, childish, irrational rituals, born out of fear of the unknown. But how much does it cost to knock on wood? 😁
FinsRule
FinsRule
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August 9th, 2010 at 5:02:02 AM permalink
He's opening the door because it's a TV show, and that's how the show worked.

Sorry, that sounds mean. I've read your post a couple of times, and I don't quite understand the point. Can you please explain why it matters if we say he "Had" to or "Chose" to. Maybe I'm just a numbers person. Which isn't good because I'm in HR.
DJTeddyBear
DJTeddyBear
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August 9th, 2010 at 5:59:56 AM permalink
What he means by had/chose is this: If you picked a goat, there was only one door left with a goat, so he HAD to open that one. If you chose the car, Monty could open either door.

Bear in mind that on the TV show, the 'doors' were actually curtains or boxes. Only the final deal used doors, and in that deal, two contestants chose doors, so there was no option for Monty to let you change your mind. The thing where Monty gave a choice was when all three curtains/boxes were in play. And it was actually rare for two curtains/boxes to contain a goat/zonk. Sometimes he opened a curtain/box to show an intermediate level prize before giving the option to switch.

And he didn't always give the option to switch.



That said, in what has become known as "The Monty Hall Paradox", three doors are used, one has a car, two contain goats, and after picking, you are ALWAYS shown a goat behind one of the other doors, and offered the option to switch.

For more info: http://en.wikipedia.org/wiki/Monty_Hall_problem
I invented a few casino games. Info: http://www.DaveMillerGaming.com/ 覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧 Superstitions are silly, childish, irrational rituals, born out of fear of the unknown. But how much does it cost to knock on wood? 😁
odiousgambit
odiousgambit
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August 9th, 2010 at 6:16:42 AM permalink
To fully comprehend it, I had to see the Wizard's example. In particular the example that follows "if the player had a strategy of switching."

http://wizardofodds.com/askthewizard/122
the next time Dame Fortune toys with your heart, your soul and your wallet, raise your glass and praise her thus: 典hanks for nothing, you cold-hearted, evil, damnable, nefarious, low-life, malicious monster from Hell! She is, after all, stone deaf. ... Arnold Snyder
konceptum
konceptum
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August 9th, 2010 at 10:30:28 AM permalink
The problem I've had with the Monty Hall paradox is the assumption that I want a car instead of a goat. Quite frankly, a goat makes a pretty good pet.
cclub79
cclub79
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August 9th, 2010 at 11:41:39 AM permalink
Quote: konceptum

The problem I've had with the Monty Hall paradox is the assumption that I want a car instead of a goat. Quite frankly, a goat makes a pretty good pet.



Better gas mileage and produces meat and milk rather than simply consuming resources.
mkl654321
mkl654321
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August 9th, 2010 at 2:43:49 PM permalink
Quote: konceptum

The problem I've had with the Monty Hall paradox is the assumption that I want a car instead of a goat. Quite frankly, a goat makes a pretty good pet.



I've had some cars that I would have preferred a goat to, all American cars made in the low-quality 80's, when Ford, GM, etc. knew that there were so many people that would "BUY AMERICAN GOLDURNIT", that they could make their cars out of cardboard and tin and millions of people would still buy them.

And to further what you said, I never got a drop of milk out of my Ford Pinto. And on the flip side, if somebody hits your goat from the rear, the goat doesn't explode in a fireball; it just kicks whoever hit him. Hmmmm. And we all used to laugh at the guy who got the goat as his prize.
The fact that a believer is happier than a skeptic is no more to the point than the fact that a drunken man is happier than a sober one. The happiness of credulity is a cheap and dangerous quality.---George Bernard Shaw
mrjjj
mrjjj
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July 23rd, 2011 at 10:46:29 PM permalink
I know this thread is old, sorry. Its always a favorite subject of mine. I have read on other boards that the human factor means NOTHING, not sure if thats true or not?

Lets say you chose #2 and #1 is opened by a monkey (lol), not Monty, NO CAR. You are allowed to switch if you want to. Are you suppose to switch to #3? I still dont get it?


Ken
MathExtremist
MathExtremist
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July 23rd, 2011 at 11:04:17 PM permalink
If the monkey chooses randomly (and happens not to show you a car), switching or not doesn't matter. But if the monkey acts like Monty -- that is, it will always show you the goat because it knows where the car is and, under the rules, can't show it to you -- then you should always switch.
"In my own case, when it seemed to me after a long illness that death was close at hand, I found no little solace in playing constantly at dice." -- Girolamo Cardano, 1563

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