Monte Hall Problem

Have you heard of the famous Monte Hall problem in statistics? It’s freaky. And I believe it offers the best evidence that our reality is subjective.

The set up is this. Game show host Monte Hall offers you three doors. One has a car behind it, which will be your prize if you guess that door. The other two doors have goats. In other words, you have a 1/3 chance of getting the car.

You pick a door, but before it is opened to reveal what is behind it, Monte opens one of the doors you did NOT choose, which he knows has a goat behind it. And he asks if you want to stick with your first choice or move to the other closed door. One of those two doors has a car behind it. Monte knows which one but you don’t.

The trick here is that most people assume it makes no difference if they stick with their original choice or move to the other door. They believe the odds are 50% either way, since there are only two choices and you don’t know anything about either choice. But mathematicians say that is wrong. You substantially increase your odds by switching doors.

That is interesting enough on its own. (I’ll give a link later that explains the math of it.) But here is the freaky part. You only improve your odds by switching doors if Monte Hall knows what is behind each door. If he simply got lucky and opened a door with a goat behind it, your odds are unchanged. In other words, your odds are changed by Monte’s knowledge, and your knowledge that Monte has that knowledge.

If reality were objective, statistics wouldn’t be influenced by knowledge. That means your world is either partly created by your mind, or you are a hologram created by some other mind, and there are a few bugs in the software.

Here’s a link to more than you want to know about the Monte Hall problem.

http://en.wikipedia.org/wiki/Monty_Hall_problem

You see the same sort of thing happen in the classic double-slit experiment in physics. The result of the experiment changes if the observer has additional information about what slit a photon passes through. Again, knowledge changes the real world. That can’t happen in the world you imagine you are living in. It has to be a bug in the hologram program. At the very least it shows that your reality is subjective.

Here’s more on the double-slit experiment.

http://en.wikipedia.org/wiki/Double-slit_experiment

DMD

Comments

[Sarah]

This is really great, is this column still in print in one newspaper?

Sarah
Note Promote Editor
Http://www.NotePromote.com

[Stephen Young]

I have written a short JavaScript program that simulates the Monty Hall game. Click on a door to select it and then a goat will be revealed behind another door. You should then click on Swap or No Swap. The result is then counted according to whether you won or lost and whether you swapped or not. You can save yourself a lot of clicking by playing 1000 games at a go by selecting "1000 Plays Swap" or "1000 Plays No-Swap"
The page is at http://members.chello.at/stephen.joung/monte.html

[Chris L]

I know I'm getting in late on this, but I feel the need to clarify something. (I was going to weigh in on the Monte Hall problem, but that has been discussed to death, and I have no idea where your mind is on it now, ditto for the double slit, but I'll be brief.).

The reason given by modern physics for the phenomenon in the double-slit experiment that you discuss is the following.

As the electron is travelling through the vacuum towards and after the slit, it does not exist in a particular place. If you observe it as it goes through the slit, then you interact with it and fix it in a position in space and time. It's not the knowledge of which slit it goes through that changes things, it's the interaction that is caused by the act of observing it. If nothing interacts with the particle, then it truly does go through both slits. Whereas if there happens to be some dust in the way of the slits, then even though we still don't know which slit it went through, it will have interacted with the dust and "picked" a slit.....ok this is turning out to be not so brief, so I'll just end it there. I'd love to continue this discussion, but not with posts in a forum.

[HGB]

Most people have said that your statement

"You only improve your odds by switching doors if Monte Hall knows what is behind each door. If he simply got lucky and opened a door with a goat behind it, your odds are unchanged."

is false, and that it doesn't matter whether Monty knows what's behind the doors or just gets lucky and happens to open a door with a goat behind it.

In fact, you are correct, and it illustrates the point I was trying to bring up with my earlier example.

What people consistently miss is that, although there is a 1/3 chance that Monty will open the door with the car, this probability is -dependent- on whether you initially choose the correct door or not. Clearly, if you initially choose the door with the car, then regardless of what remaining door Monty chooses, the game will continue.

However, if you initially choose the wrong door, then there's a 1/2 chance that the game will end because Monty revealed the car.

The critical thing to realize is that, once you know that Monty randomly choose a door and picked one with a goat, this provides evidence to decrease the probability that you initially picked the wrong door. In fact, there's a 1/3 chance you picked the correct door, and there's a 2/3 chance that you picked the incorrect door -multiplied- by a 1/2 chance that Monty also picked the incorrect door, which gives you 1/3 as well. (The other 1/3 is when Monty picks the correct door and the game ends.)

Thus, you have a 1/2 chance either way.

This is related to what I was driving at with my earlier example:

Suppose there's a disease that affects 1/1000 of the population. A test has been developed to determine if you have the disease, that is 99% accurate, that is to say: if you have the disease, then the test will come back positive 99% of the time. If you don't have the disease, then the test will come back negative 99% of the time.

You want to know if you have the disease, so you go get the test and it comes back positive. What's the probability that you have the disease?

Most people will say, given that the disease is 99% accuracy, and it came back positive, you have a 99% chance of having the disease.

However, this is false. Formally, you need to use Bayes Rule to do the computation. Intuitively, the explanation is that, since the disease only affects 1/1000 people, there's a much higher probability that you were one of the 999/1000 people who are not affected and got a false positive with probability 1/100, then you were one of the 1/1000 people with the disease and got a true positive with probability 99/100. Do the math, and you'll find that, given that the test came back positive, there's about a 10% chance that you have the disease, not 99%.

[Piache]

There is one thing wrong on your description.

"You only improve your odds by switching doors if Monte Hall knows what is behind each door. If he simply got lucky and opened a door with a goat behind it, your odds are unchanged."

This is not true.

If he simply got lucky and opened a door with a goat behind it, your odds changes in the same way.

Your odds changes not because Monte Halls knows what is behind each door, they change because YOU know one of the doors does not have the prize.

It doesn't even matter if there is someone opening doors.

Suppose that you pick a door, then a computer randomly picks one of the remaining doors and shows you the content.

If the door has a goat, then if you change doors you increase your odds.

If the door has a prize, then your odds also changes: the odds of winning are 0.

[Amun]

Using the the uncertainty principle to claim that the universe changes depending on what you know of it and is therefore subjective has always bothered me. In our existance, we cannot know something without observation. That is a limitation of what we are. If we were some kind of creature that could view the path of photons without disturbing them then this argument would never have existed.

[Harry Johnston]

Dave: quantum entanglement definitely does *not* allow faster than light communication. That is, you can't actually send your "morse code" faster than light. This can be demonstrated in the general case by choosing an appropriate formulation of quantum mechanics, but you can also take any specific proposed design and if you do the maths properly you find that there is no way for an experimenter at the "receiving end" to determine the setting at the "transmitting end". You can't even make a better than random guess.

In the specific design you describe the "receiving end" will always be uniformly lit (an equal probability of a photon arriving at any point) regardless of the layout of slits at the "transmitting end". The only correlation you'd get between the two ends is that when a photon hits the screen at the transmitting end (as opposed to missing the slits) it will have landed in one of two specific areas of the screen at the receiving end - note that I'm assuming that the initial direction of travel is the entangled property.

People often say that quantum entanglement is weird, but once you've really got your head around quantum mechanics, quantum entanglement is no stranger than classical correlations. (It's only the presence of non-commuting observables that makes the Bell experiment so counter-intuitive.)

Imagine having one red and one white marble in a bag, taking one out at random, and sending it to Alpha Centauri by special delivery without ever looking at it. Then look in the bag. If you see a red marble, you instantly know that the white marble is at Alpha Centauri - spooky action at a distance?

(For what it's worth, I'm not a professional physicist, but I did complete a PhD in quantum optics.)

[Kristina L.]

I think this proves that statistics are subjective more than that reality is. I've been a little suspicious of both statistics and proof by induction for quite a while.

[Douglas]

An example of how knowledge changes probability:

Say you have two cards, one with "this is the winning card" written on it, and one with "this is the loosing card" on it. To win, you have to pick the winning card. People who can read will win every time, they have a 100% chance of winning. A blind person, who cannot read the cards, will have to pick at random. They will win 50% of the time.

This is the same as in the Monty Hall problem, Monty knows which is the winning door and will always leave it for you to pick.

[Matt McNamara]

The easy way to rationalize "The Monte Hall Problem" is to think of it the other way...

One door has a prize. Two do not. You have a 1 in 3 chance of picking the correct door. However, you have a 2 in 3 chance of picking the WRONG door. By switching your answer after Monte's reveal, you swap your odds.

[Joachim Dyndale]

I believe this is an example of a purely theoretical problem. In reality, one might as well ask Monty to remove a goat before the game begins - giving a 50% chance of choosing the car.

Practically, there is no difference between the two scenarios for the contestant.

[E]

Gordon goosemonster wrote:

[Not really anything to complex here.

You initialy have a 1/3 chance of wining.
You are then shown a goat.
Your odds change to 1/2 as you now have the option to choose again. You don't actually have to change your door, choosing to keep your original selection still gives you a 1/2 chance.
Any other way of looking at it is just typical American stupidness.]

You'll have more credibility about "typical American stupidness" if you're less of a dumbass yourself.

It is counterintuitive.

But my God, there are links with pictures to explain it. Even someone like me, a typical stupid American... who went to public school... can get it.

Initially you have a 2/3 shot at picking a goat.

On pick two, if you picked the car first and switch, you'll lose 1/3 of the time.

THEREFORE, if you did Not pick the car first and switch, you'll Not lose 2/3 of the time.

That is not even money.

If you miss the car on your first guess, the reveal combined with the switch only leaves winning as a possible outcome. So your chance to win becomes the same as your initial chance to lose.

And if you look carefully, you'll see Monte Hall, with the score so far...

Typical Stupid American: 1
Arrogant Foreign Goat-Fucker: 0

Would you like to switch doors?

[stefan]

It is unlikely that the one door you pick has the car. That's the 1/3 thing. And yet the car could in reality be there the one time that you play the game.

The probability is worked out by considering multiple plays. But if you only play once, as a contestant, as implied by the game show scenario, then it makes no difference switching.

I think this is why many people get the answer "wrong" from a mathematical point of view, but are right in practice.

It's not a difference between object and subject, it's a difference between multiple and single plays.

The odds being against you doesn't prevent you from winning in reality. Every week some guy wins the lottery.

[Matt]

The perspective of the contestant is subjective, but the position of the car is objective.

[James F. R. Wright]

The idea that "your odds are changed depending on Monte's knowledge" is freaky and blows your mind shouldn't do and isn't since it's a misnomer. The set up where the presenter knows which curtains the goats are behind is a completely different set up to the one where he doesn't know. So each set up has its own probability.

[marco polo]

I used to think you were crazy-smart and eloquent.

Now I'm not so sure, maybe your spectacular presentation and humor bamboozle me most of the time, and you're secretly diluting my IQ.

Please post again on this topic, to acknowledge you didn't have your coffee today, or at least try making your point again? I mean jeez

[John]

Scott: "In other words, your odds are changed by Monte’s knowledge, and your knowledge that Monte has that knowledge."

The odds are set by what the question states that Monty will do. Your speculation about his knowledge is specious; he could be a robot and the odds would be the same.

The knowledge of the experimentor ("you") affects your assessement of the odds. But not the odds themselves - they don't depend on what the experimentor knows.

Quantum indeterminacy is interesting and wierd precisely because it is fundamentally different from conventional causal probability. If you can't see the difference that doesn't mean you've discovered something. It means you've failed to "get" what Bohr, Heisenberg and others did discover.

[John]

Scott: "In other words, your odds are changed by Monte’s knowledge, and your knowledge that Monte has that knowledge."

The odds are set by what the question states that Monty will do. Your speculation about his knowledge is specious; he could be a robot and the odds would be the same.

The knowledge of the experimentor ("you") affects your assessement of the odds. But not the odds themselves - they don't depend on what the experimentor knows.

Quantum indeterminacy is interesting and wierd precisely because it is fundamentally different from conventional causal probability. If you can't see the difference that doesn't mean you've discovered something. It means you've failed to "get" what Bohr, Heisenberg and others did discover.

[Greg]

You show promise in acknowledging that reality is a subjective experience, but show cartoon logic in some of your other conclusions.

Since I've argued with other people about this subject in the past I've put my response on my blog @ http://gregbecerra.blogspot.com/2008/04/mysticism-of-monty-hall.html

Times like this makes me wonder if you draw Dilbert or if Dilbert draws you.

[Chris]

All this talk about Monty having or not having any knowledge is silly and beside the point. Monty's knowledge is irrelevant. What is important is *your* knowledge. If your level of knowledge increases then your odds change; it doesn't matter if that knowledge comes from Monty himself or from knowing what is behind the door he picks.

As to the double slit experiment, anyone who's interested in understanding the double-slit experiment and modern quantum mechanics *needs* to read the essays here: http://www.overcomingbias.com/science/index.html. Some other posters have mentioned that link but it deserves to be reiterated. Scott, if you haven't read that yet then you definitely should.

[addendum to common sense]

Think of it like this: If a third party enters AFTER the host has already revealed a goat and doesn't know which door you originially picked, his odds are 1/2 to 1/2. You, though, have prior knowledge, so your odds are 2/3 to 1/3.

Scott, I think your problem is in your definition of probability. Probability DOES NOT describe reality; it quantifies the possibilities of random events. The more knowledge you have, the less unknown (which, some might argue, is behind our perception of randomness) the future is, and the higher probability you have of being right.

[common sense]

This is actually very simple.
Originally, you have a 1/3 chance of being correct. Once you choose a door, it doesnt matter what happens around you anymore - a nuclear explosion could destroy all life for all you care - your odds of winning are still 1/3. YOUR DOOR will only win 33% of the time.
So, when the host opens one of the other doors that contains a goat, that doesn't affect your original probability - the door you opened originally will still only win 33% of the time! Since there's only one door left, it HAS to win the other 66%!
My theory is an extension of the following scenario: Suppose you chose a door (we'll call this group A), and then (without opening a door) the host asks you if you want to switch. Well, your probability of winning is 33% now, so it should seem logical to switch doors, because 66% of the time the car will be behind one of the other doors (we'll call these doors group B), no? The thing is that group B contains two doors, so any INDIVIDUAL door you choose will always have a probability of 33%. You can't improve your chances by switching.
But if the host already revealed one losing door, switching works out for you, because he took a door out of group B! Now, group B, which has a 66% chance of winning, only contains one door!
The math works out even assuming that reality is real.

[Martin Bucknall]

My chances of winning aren't 1/3 but 2/3. You can get about in Edinburgh using a goat, but using a car it's impossible.

[yes]

lol "That is interesting enough on its own. (I’ll give a link later that explains the math of it.) But here is the freaky part. You only improve your odds by switching doors if Monte Hall knows what is behind each door. If he simply got lucky and opened a door with a goat behind it, your odds are unchanged. In other words, your odds are changed by Monte’s knowledge, and your knowledge that Monte has that knowledge.

If reality were objective, statistics wouldn’t be influenced by knowledge. That means your world is either partly created by your mind, or you are a hologram created by some other mind, and there are a few bugs in the software."

biggest load of crap i've ever seen. i know you're smart (you're in mensa for goodness sake) so if u do believe this it's just confirmation bias.

basic probability: Pr(A) is usually different to Pr(A given B). it's got nothing to do with holograms, and continuing to say something doesn't cause it to be true. stop trying to hypnotise people.

[Gordon goosemonster]

Not really anything to complex here.

You initialy have a 1/3 chance of wining.
You are then shown a goat.
Your odds change to 1/2 as you now have the option to choose again. You don't actually have to change your door, choosing to keep your original selection still gives you a 1/2 chance.
Any other way of looking at it is just typical American stupidness.