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The Monty Hall problem... king of puzzling puzzles...

PosTMOd

Well-Known TRIBEr
You are on a game show. There are three doors in front of you. Behind two doors are goats, and behind one door is a fabulous prize.

You choose a door.

The game show host then opens up a door to reveal... goats.

Now, is it better for you to stay with your first choice, or should you switch?

The extremely counter-intuitive answer: IT IS BETTER TO SWITCH. The odds of winning if you stay with the original = 1/3
The odds of winning if you switch: 2/3

Neat, eh?
 

AshG

Member
not true.
the odds of you winning depends on your initial guess, and your ability to choose based on the additional information provided by monty revealing the contents of door x.
assuming you picked a non-goat door, then if you were to switch your choice at this point in the game, your odds of winning are now 1/2 (assuming there is 1 correct solution and 2 goat solutions in total).
Still 'tis better than the 1/3 chance you'd have by sticking with your initial guess, but only 50% better, rather than 100% better chances (1/2 vs. 2/3, compared to 1/3).
 
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PosTMOd

Well-Known TRIBEr
Originally posted by AshG
not true.
the odds of you winning depends on your initial guess, and your ability to choose based on the additional information provided by monty revealing the contents of door x.
assuming you picked a non-goat door, then if you were to switch your choice at this point in the game, your odds of winning are now 1/2 (assuming there is 1 correct solution and 2 goat solutions in total).
Still 'tis better than the 1/3 chance you'd have by sticking with your initial guess, but only 50% better, rather than 100% better chances (1/2 vs. 2/3, compared to 1/3).
Wrong. That's why it's counterintuitive.

But, try again.

Starts out: 1/3 for winning each door, and 2/3 NOT winning from the other two doors combined. Since the door that is shown to you is one of the NOT winners, the odds of the door you didn't pick become the whole 2/3.

Here, this will explain it to you better... it's actually one of the most interesting things on the planet for being so counterintuitive:

http://www.io.com/~kmellis/monty.html
 

Subsonic Chronic

TRIBE Member
I saw that on another website somewhere too, about how your chances are better if you change your decision. I had to read it over and over before it finally made sense to me. And it still doesn't totally because it defies simple logic.

Pete
 

defazman

TRIBE Member
simple

1 non-goat prize for 3 doors = 1/3 chance of picking non-goat

if you chose an incorrect door, then with one door known:

1 non-goat prize for 2 doors = 1/2 chance of picking non-goat

now I'm not really sure if I understand this though. When you switch, do the prizes get switched around behind the doors again? Is it still 3 unkowns, or does the door you chose remain open?
 

Rocky

TRIBE Member
well the way i see it is that it really doesn't matter if you pick a door initially (unless the one that you pick is in fact the winning door). when all is said and done, one door that doesn't contain the prize is eliminated and 2 doors remain. therefore, you have a 50% chance of choosing the winning door.

let's say that you do initially choose a door. it really doesn't matter which one you choose (unless it's the winning door of course) because one non-winning door is going to be eliminated. when that door is eliminated you can do one of two things. you can either stay with the door you have chosen or switch to the other door. that's 1/2 choices and...let me get my calculator.....

.....1/2 equals 50%

...hmmm, what's is wrong with this analysis? i can't seem to find anything wrong with it.

...although, since i have a lot of time on my hands, i did a test to see which theory is correct. i played the game several times and switched "doors" everytime. out of 19 games, i won 14 by switching. that is 73.7%. if i would have stayed with my initial door the chances of winning would have been 26.3%. that is very close to the theoretical 2/3 switch and 1/3 stay.
 
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bitchass

TRIBE Member
http://www.comedia.com/hot/monty-answer.html
Answer to the Monty Hall Problem
Hold on to your hats...
you *double* your chances by switching
This is, at first look, way counter-intuitive, so here's an attempt at an explanation:

Take a look at this matrix of possibilities:

Door
~~~~
case A B C
~~~~
1 bad bad good
2 bad good bad
3 good bad bad

Let's assume you choose door A -- you have a 1/3 chance of a good prize.

But (this is key) Monty knows what is behind each door, and shows a bad one.

In cases 1 and 2, he eliminates doors B and C respectively (which happen to be the only remaining bad door) so a good door is left: SWITCH!

Only in case 3 (you lucked out in your original 1 in 3 chances) does switching hurt you.

So, your probability goes up from 1/3 to 2/3 if you switch after being shown a bad door.
 

PosTMOd

Well-Known TRIBEr
^^^ BS.

It really boils down to the fact that information is passed... that information being that the one door being eliminated MUST ALWAYS be one with a goat, not the prize.
 
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Adam

TRIBE Member
If 'Package A' is your choice of door, you have a 1 in 3 chance. If 'Package B' is the combination of both of the doors you didn't choose, there is a 2 in 3 chance it contains the car. You are obviously wiser chosing Package B, and by Monty narrowing down which door in Package B contains the car, you're increasing your odds to 2 out of 3.
 

PosTMOd

Well-Known TRIBEr
It's a psychological test to see if you are merely contrary for the sake of it, or you can actually think for yourself.

Until your frontal lobes have matured to a certain extent, you will always argue with the counterintuitive bit. Afterwards... well, you'll see.
 

PosTMOd

Well-Known TRIBEr
I'm reminded of something my eldest sister would do to me when I was 2 or 3 years old; she would pour two glasses of orange juice, one a tall and skinny glass, one a short and fat glass. She would put way less in the tall and skinny one, but I would always choose it regardless, since it seemed to me to have more in it, since the level of juice would be higher in it. She would laugh and laugh and I never could understand why, for I absolutely *knew* I was getting more juice than her.
 

Shug

TRIBE Member
PosTMOd said:
It's a psychological test to see if you are merely contrary for the sake of it, or you can actually think for yourself.

Until your frontal lobes have matured to a certain extent, you will always argue with the counterintuitive bit. Afterwards... well, you'll see.
I had to write out the full decision tree before I wrapped my head around it. The point most people miss is that you choose first, and the probability is measured from that instance, before the first goat is shown. I wouldn't say it's actually counter-intuitive, I would say the problem is laid out in such a way that it leads you down the false path.
 
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lok

TRIBE Member
Shug said:
I had to write out the full decision tree before I wrapped my head around it. The point most people miss is that you choose first, and the probability is measured from that instance, before the first goat is shown. I wouldn't say it's actually counter-intuitive, I would say the problem is laid out in such a way that it leads you down the false path.
This is a language "problem" not a logical one. Just like that "math" puzzle where you split money and end up with $27 from $30
 

Bass-Invader

TRIBE Member
unless the fabulous prize is also goats, which is a pretty good prize, especially if they are the goats which were genetically modified with spider genes to produce spider silk milk.
 
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