The Monty Hall Problem and Schrodinger's Cat

Mar 28, 2013 General Nonsense |

The famous Monty Hall problem in the field of statistics goes like this: Monty Hall is a game show host. You are given a choice of three doors. One has a car behind it, the other two have goats. If you pick the door with the car, you win it. Your odds are 1-in-3.

So you pick a door, but before it opens, Monty opens one of the other two doors to reveal a goat. He asks if you want to switch from the door you initially picked to the other closed door. Your brain says the odds are the same for any closed door, so you stay. But in fact, the odds are twice as good if you switch doors.

You can see the math of it here. But if you are normal, you'll never reconcile in your mind how one closed door could have better odds than the other. If there are two closed doors remaining, how can the odds be anything but 50-50?

This reminds me of the Schrodinger's Cat thought experiment in which a cat in a sealed box (presumably with air holes) exists in a state of being simultaneously alive and dead depending on the results of a randomized event happening inside the box. How can a cat be alive and dead at the same time? Math says it can happen, my brain says no.

The pattern recognition part of my brain is connecting the Monty Hall problem with the Shrodinger's Cat thought experiment because both situations feel like proof that our brains are not equipped to understand reality at its most basic level.

Most of us accept the idea that math is a better indicator of truth than our buggy personal perceptions. Math doesn't lie, but our brains are huge scam artists. The Monty Hall problem and Schrodinger's Cat are examples in which our perceptions of reality and the math of reality disagree in a big way. It makes me wonder how much of the rest of my so-called reality disagrees with math without me knowing.

If I were programming a computer simulation full of artificial humans who believe they are real, I would need to take some shortcuts in creating their context and history. It would be nearly impossible to invent consistent histories for seven billion people spanning back to the primordial ooze. A far smarter approach would be to craft the history as you go, based on the present, in whatever minimum way is necessary to make all histories consistent.

For example, let's say you learn that you are the grand winner of a lottery. At the moment you realize you are the big winner, history becomes limited to only the possibilities that got you to that winning moment. Before you learned you were a winner, the reality at the lottery headquarters was only a smear of possibilities - like Shrodinger's Cat - where you were both a winner and a loser, just like everyone else. As soon as you learn you won, your history and everyone else's harden to conform to it. No one else can perceive that they won the grand prize in that particular lottery.

If I were the programmer of this simulation that you call your reality, I would make the history dependent on the present just to streamline my work. All I need from my fake history is that it is consistent with all the other fake histories so there is no "tell" left by the programmer.

I realize the simpler explanation for my confusion about Monty Hall and Schrodinger's cat is "Math be hard." But I like the psychological freedom of feeling as though I am the author of my own history and not its bitch.

Here's the cool part: I get to keep my interpretation of reality - in which my history is a manufactured illusion - until something in my present experience is inconsistent with that view.

Recently I heard of two senior citizens with mild dementia who became friendly at a senior care facility. Their fragile minds concocted an elaborate history of being childhood acquaintances that had found each other through fate. No one tries to dissuade them of this illusion because it works for them. They successfully rewrote their histories without any repercussions.

I wonder how often the rest of us rewrite our histories. Our only limitations are that our new histories have to be consistent with whatever scraps of history have already hardened.

So you pick a door, but before it opens, Monty opens one of the other two doors to reveal a goat. He asks if you want to switch from the door you initially picked to the other closed door. Your brain says the odds are the same for any closed door, so you stay. But in fact, the odds are twice as good if you switch doors.

You can see the math of it here. But if you are normal, you'll never reconcile in your mind how one closed door could have better odds than the other. If there are two closed doors remaining, how can the odds be anything but 50-50?

This reminds me of the Schrodinger's Cat thought experiment in which a cat in a sealed box (presumably with air holes) exists in a state of being simultaneously alive and dead depending on the results of a randomized event happening inside the box. How can a cat be alive and dead at the same time? Math says it can happen, my brain says no.

The pattern recognition part of my brain is connecting the Monty Hall problem with the Shrodinger's Cat thought experiment because both situations feel like proof that our brains are not equipped to understand reality at its most basic level.

Most of us accept the idea that math is a better indicator of truth than our buggy personal perceptions. Math doesn't lie, but our brains are huge scam artists. The Monty Hall problem and Schrodinger's Cat are examples in which our perceptions of reality and the math of reality disagree in a big way. It makes me wonder how much of the rest of my so-called reality disagrees with math without me knowing.

If I were programming a computer simulation full of artificial humans who believe they are real, I would need to take some shortcuts in creating their context and history. It would be nearly impossible to invent consistent histories for seven billion people spanning back to the primordial ooze. A far smarter approach would be to craft the history as you go, based on the present, in whatever minimum way is necessary to make all histories consistent.

For example, let's say you learn that you are the grand winner of a lottery. At the moment you realize you are the big winner, history becomes limited to only the possibilities that got you to that winning moment. Before you learned you were a winner, the reality at the lottery headquarters was only a smear of possibilities - like Shrodinger's Cat - where you were both a winner and a loser, just like everyone else. As soon as you learn you won, your history and everyone else's harden to conform to it. No one else can perceive that they won the grand prize in that particular lottery.

If I were the programmer of this simulation that you call your reality, I would make the history dependent on the present just to streamline my work. All I need from my fake history is that it is consistent with all the other fake histories so there is no "tell" left by the programmer.

I realize the simpler explanation for my confusion about Monty Hall and Schrodinger's cat is "Math be hard." But I like the psychological freedom of feeling as though I am the author of my own history and not its bitch.

Here's the cool part: I get to keep my interpretation of reality - in which my history is a manufactured illusion - until something in my present experience is inconsistent with that view.

Recently I heard of two senior citizens with mild dementia who became friendly at a senior care facility. Their fragile minds concocted an elaborate history of being childhood acquaintances that had found each other through fate. No one tries to dissuade them of this illusion because it works for them. They successfully rewrote their histories without any repercussions.

I wonder how often the rest of us rewrite our histories. Our only limitations are that our new histories have to be consistent with whatever scraps of history have already hardened.

Apr 3, 2013

Here's a great paper explaining the real-world game dev application for what you describe:http://my.safaribooksonline.com/book/programming/game-programming/9781584507024/ai-level-of-detail-for-really-large-worlds/ch20lev1sec1

"AI Level of Detail for Really Large Worlds"

Basically, if there's no player "on grid" to observe the simulation directly, then you can simulate aggregates with statistics.

The example given is a pub if a village with NPC (AI players) drinking in the pub, which is fun.

Apr 2, 2013

Late as usual. So, anyway, I love the visualization concept. re: Monty Hall. Imagine 3 garage doors in front of contestant 1. One door is on a 1 car garage. The other 2 doors are on a 2 car garage. Monty gives the spiel, blah blah. But he says you can pick the garage. and get to keep the car if it's anywhere in the garage you pick. What garage will you pick? Obviously the 2 door garage. Now imagine a space between the two doors equal to the space between the original two garages. But contestant 2 didn't see them separate the 2 door garage. 2 picks one of the now single door garages (all 3 looking the 'same' now). Monty then merges the other two doors back into a single garage with 2 doors and offers you to keep the garage you have or switch to the double door, getting to keep the car regardless of which garage. You obviously switch to the double door garage. Whether there's a goat behind one of the 2 doors or not is irrelevant. Because...contestant 2 had only a 1 in 3 chance the first time, now he has a 2 in 3 chance. By showing contestant 2 the goat, he's tricking contestant 2 into thinking that the odds changed. They didn't. But, remember, this is my take on reality. Yours may be different
Apr 1, 2013

[On the Monty Hall question, your explanation is by far the best I have seen. Your case is airtight. The problem is that it is also an airtight case to say that if there are two doors left, and you have no information about them, the odds of the car being behind a particular one has to be 50%. The third door that is opened to reveal a goat cannot transfer its odds to a different door by magic. It can only remove itself from the game. So while your case is completely logical, so is the opposite case. It looks like a programming bug in our simulation. -- Scott]You DO have information about those doors after goat #1 is revealed: You know each either hides goat #2 or a car. The odds are only 50% if there is one car, one goat, and two doors. You can't ever arrive at a "50%" number without ignoring one of the 3 doors. Which is illogical in a problem consisting of 3 doors. The incorrect view simply isn't logical no matter how much your illogical meat-brain thinks it is.

Apr 1, 2013

Isaac Asimov used to write that he got smarter and better looking as he got older. Whether this was objectively true or not really wouldn't have mattered to him - and that's the point. Smart people invent the future they live in by bending the rules of the reality they live with. As a science fiction writer Asimov was concerned with both and he was happier for it.Science fiction has played wonderful games with the idea of history - from Blade Runner where androids were given implanted memories that never happened to them, to The Matrix where all of history and collectively experienced reality was nothing but fiction. It is interesting to ponder whether we'd be better off if memory were like a computer hard drive that could be rewritten with new data that is better suited to the present. But if that were possible, would we even be human any more? I don't think so. And for me, it is a happier thought to think about the good of being human, than the good of being something else.

And while I am thinking that, I will get infinitesimally smarter, and better looking, as the time passes. :)

Mar 31, 2013

Scott, come on. Is that really ("And you know that how?") your defense of the proposition that reality isn't reality? Because you don't accept that reality exists, you can then say that with certainty that reality doesn't exist?That is the classical circular argument. Of course, I can't ask you to prove that reality doesn't exist, because you can't prove a negative. But if you believe that reality doesn't exist, then give some proof other than saying, "Joe doesn't believe in reality. Therefore, there is no reality."

That's specious, and a wasted argument. You can do better.

Mar 30, 2013

{I wonder how often the rest of us rewrite our histories. Our only limitations are that our new histories have to be consistent with whatever scraps of history have already hardened.} - ScottWasn't that the basic premise of 1984? That history was getting rewritten constantly to fit modern needs.

Mar 30, 2013

Actually the monty hall problem is about an improper perspective.Logically speaking:

There are 3 goats.

You know one of the doors you don't pick has to be a goat.

You are never told if your door has a goat or not.

In short, revealing the goat is useless; you knew it was there anyways and frankly you don't give a darn about it. The problem is the goat screws up your perspective.

Here is a proper perspective: 2 out of 3 times, you do not pick a goat with your first pick. You only get it right on your first pick 1 out of 3 times. Therefore, 2 out of 3 times, switching is in your benefit.

Mar 30, 2013

If I were the programmer of this simulation that you call your reality, I'd make my job easier by programming 6.5 billion NPC (non-playing characters). They don't need any history unless they interact with a playing character. Most of the NPCs will simply appear as statistics on the news: 1,000,000 dead from disease, another 10,000,000 from starvation, 250,000 in an earthquake, etc.
Mar 30, 2013

@AtlantaDudeIf you want to understand why, all you have to do is read my last post. :-)

Mar 30, 2013

OK. I have finally come around on this. I ran my own simulation, and switching was better. Still couldn't understand why. Then I looked at the inverse - what if I was a dedicated switcher and I wanted to lose. When Monty reveals a goat or mule, he takes away one of my chances to lose, so he made it harder for me to lose, and thus easier for me to win.And, despite all that, I still don't really understand "why" this is the answer and my previous analysis is wrong. The only thing I can say with certainty is that I remember why I always hated probability classes.

Mar 29, 2013

To simplify further, let's just look at the switcher scenarios1. Pick-Car, Reveal-Goat: You Switch to Mule - Lose

2. Pick-Car, Reveal-Mule: You Switch to Goat - Lose

3. Pick-Goat, Reveal-Mule: You Switch to Car - Win

4. Pick-Mule, Reveal-Goat: You Switch to Car - Win

5. Pick-Goat, Reveal-Car: Game Over - Lose

6. Pick Goat, Reveal Goat: Game Over - Lose

7. Pick-Mule, Reveal-Car: Game Over - Lose

8. Pick Mule, Reveal Mule: Game Over - Lose

9. Pick Car, Reveal Car: Game Over - Win

Switcher has 3 wins in 9 scenarios. Or, if you want to just concentrate on the situations in which Monty opens a non-chosen losing door - that is scenarios 1 thru 4 and Switcher has 2 wins in those scenarios.

Now let's look at the possibilities just for a person who keeps his original choice

1. Pick-Car, Reveal-Goat: You Keep Car - Win

2. Pick-Car, Reveal-Mule: You Keep Car- Win

3. Pick-Goat, Reveal-Mule: You Keep Goat - Lose

4. Pick-Mule, Reveal-Goat: You Keep Mule - Lose

5. Pick-Goat, Reveal-Car: Game Over - Lose

6. Pick Goat, Reveal Goat: Game Over - Lose

7. Pick-Mule, Reveal-Car: Game Over - Lose

8. Pick Mule, Reveal Mule: Game Over - Lose

9. Pick Car, Reveal Car: Game Over - Win

Keeper also has 3 wins in 9 scenarios. Or, if you want to just concentrate on the situations in which Monty opens a non-chosen losing door - that is scenarios 1 thru 4 and Keeper has 2 wins in those scenarios.

So the odds are the same for Keeper and Switcher, regardless of whether you consider just the cases where Monty opens a loser door, or if you look at all 9 scenarios (3 choices by you * 3 choices by Monty)

[The fascinating thing is that opposing arguments are both 100% airtight. It's only the human impulse to pick sides that makes us think there is a clear winner as far as the logic goes. But I believe the simulations on this always support the better odds of switching. -- Scott]

Mar 29, 2013

I will boil it down for the doubters. Rather than just giving me a thumbs down, please tell me what other possible scenarios exist, other than the 9 below. 1. Pick-Car, Reveal-Goat: You can either Keep Car or Switch to Mule

2. Pick-Car, Reveal-Mule: You can either Keep Car or Switch to Goat

3. Pick Car, Reveal Car: Game Over - Win

4. Pick-Goat, Reveal-Mule: You can either Keep Goat, or Switch to Car

5. Pick-Goat, Reveal-Car: Game Over - Lose

6. Pick Goat, Reveal Goat: Game Over - Lose

7. Pick-Mule, Reveal-Goat: You can either Keep Mule, or Switch to Car

8. Pick-Mule, Reveal-Car: Game Over - Lose

9. Pick Mule, Reveal Mule: Game Over - Lose

There are no other branches to this tree. There 9 scenarios. You have three chances to win if you start the game with a Keep strategy and three chances to win with a Switch strategy

Mar 29, 2013

@Melvin1 - That doesn't matter. Instead, the fact that Monty intentionally opens a goat/mule door is the reason why there is no new information revealed when he does the reveal. However, if he randomly opened another door and got a goat, then there would be Bayesian updating and the two problems - choosing before the reveal, choosing after the reveal - would not be independent.As it is, the two problems are completely independent. The first problem is like you are going to walk into one of two casinos. The first casino has a car and goat and the second casino has a car and a mule. Finding out which casino you walked into has no bearing on your odds of winning.

Mar 29, 2013

@AtlantaDude: Scott didn't explicitly say it, but the puzzle presumes that Monty intentionally opens a goat door.
Mar 29, 2013

Here is the logic explaining why switching does NOT improve your chances:The first thing to do is avoid confusion caused by there being two goats. There are two bad outcomes, and for clarity, it is better if we make one a goat and the other a mule.

Now, If we start with three equal possibilities on your first pick it can either be a goat, a mule or a car, and we assume that you have an equal chance of picking any of them. We also know that Monty will ALWAYS reveal a bad prize, regardless of what your initial pick was. Therefore, the only scenarios that exist are as follows.

Pick-Car, Reveal-Goat: Keep Car, Switch Mule

Pick-Car, Reveal-Mule: Keep Car, Switch Goat

Pick-Goat, Reveal-Mule: Keep Goat, Switch Car

Pick-Mule, Reveal-Goat: Keep Mule, Switch Car

As you can see there are two â€œKeepâ€ outcomes that result in a car, and there are two â€œSwitchâ€ outcomes that result in a car. So keeping and switching both have an equal opportunity to win, and the probability of winning is 2 out of 4, or 50%. This is the probability problem framed from the point AFTER he opens the door.

Now, what if you tell me that Monty does not ALWAYS reveal a bad prize, but that he could also reveal an unchosen door with a car in it? If that is the case, you have to add the following 2 scenarios where he reveals the car after your initial choice

Pick-Goat, Reveal-Car: Game Over-Lose

Pick-Mule, Reveal-Car: Game Over-Lose

Adding those scenarios, gives you the following possibilities

Pick-Car, Reveal-Goat: Keep Car, Switch Mule

Pick-Car, Reveal-Mule: Keep Car, Switch Goat

Pick-Goat, Reveal-Mule: Keep Goat, Switch Car

Pick-Mule, Reveal-Goat: Keep Mule, Switch Car

Pick-Goat, Reveal-Car: Game Over - Lose

Pick-Mule, Reveal-Car: Game Over - Lose

There are still two winning Keep scenarios and two winning Switch scenarios, so keeping and switching have an equal chance of winning.

Finally, letâ€™s throw in the option of Monty not doing a reveal. Instead he simply opens the door you choose initially. Those three scenarios look like this

Pick Car, Reveal Car: Game Over â€“ Win

Pick Goat, Reveal Goat: Game Over â€“ Lose

Pick Mule, Reveal Mule: Game Over â€“ Lose

Now, we can combine all these scenarios to look at every outcome. This list comprises all potential outcomes before you make your initial choice.

1. Pick-Car, Reveal-Goat: Keep Car, Switch Mule

2. Pick-Car, Reveal-Mule: Keep Car, Switch Goat

3. Pick-Goat, Reveal-Mule: Keep Goat, Switch Car

4. Pick-Mule, Reveal-Goat: Keep Mule, Switch Car

5. Pick-Goat, Reveal-Car: Game Over - Lose

6. Pick-Mule, Reveal-Car: Game Over - Lose

7. Pick Car, Reveal Car: Game Over â€“ Win

8. Pick Goat, Reveal Goat: Game Over â€“ Lose

9. Pick Mule, Reveal Mule: Game Over â€“ Lose

If you go into this game with a â€œKeepâ€ Strategy you have 3 chances to win (1, 2 and 7 above) out of 9 total for a 33% probability. If you go in with a â€œSwitchâ€ strategy, you also have three chances to win (3, 4 & 9). So there is not advantage in switching. This is the probability problem BEFORE he opens door, and you have a 33% chance of winning, regadless of whether you go in with a Keep or a Switch strategy.

So if you view the problem from the beginning, you have 33% chance with a Keep or Switch strategy. If you view the problem from the point after a bad prize is revealed, you have a 50% chance with a Keep or Switch strategy.

The confusion comes from two areas - one is mixing up or not counting all the goat outcomes, and the other is mixing up the probability as viewed from the beginning with the probability as viewed from after a door is opened.

Mar 29, 2013

It's been said a few times, but I'll jump on the bandwagon:The Monty Hall problem wasn't that hard for me to figure out -- after my initial 50:50 "obvious" answer. Assuming, of course, that Monty intentionally opens a goat door, he's giving additional information about the two doors you didn't choose, doubling those odds. No magic or reality-bending.

Schroedinger's Cat is a reductio ad absurdum showing the limits of a model (the Copenhagen interpretation) of quantum physics. His argument was that a cat is clearly NOT both alive and dead, and therein lies the weakness of applying quantum physics to the macro world.

That said, quantum physics is indeed difficult for most of us to understand - almost magical. But so were heliocentrism and gravity at one point. We've even become pretty comfortable with electricity (the most useful way of doing so for me was with plumbing analogies).

You raise an interesting question about how we rewrite histories, but you've muddied it up with incorrect examples. Those are just logic puzzles, not alternate realities.

Mar 29, 2013

I am afraid that Scott as intentionally or unintentionally bamboozled his readers with this one. The Monty Hall problem is NOT a two-thirds chance of winning if you switch. The initial problem, pick one out of three, has actually no bearing on the second problem which is pick one out of the two remaining. As soon as he opens a goat door, the original problem is over. Then you are left with a new problem, which is simply that is simply keep door / switch door, each having an equal chance of having a car.

The problem with the explanation, is that it assumes there is some dependency between the two problems, and there would be if Monty only revealed a door when you picked the car. However, he will always remove a goat door, regardless of what you originally pick. Therefore, that goat door does not "lend" its probability to the other non-chosen door. It simply reduces the denominator for both remaining doors' probability from 3 to 2.

For a better explanation see: http://ablestmage.wordpress.com/2007/11/30/the-classic-monty-hall-problem-gets-goatsed/

Mar 29, 2013

I expect that thousands of years ago most humans would have summarily rejected the idea that the Earth is a sphere that orbits the Sun because it was so completely counter-intuitive. Our intuition has evolved.
Mar 29, 2013

[Pretending, or deluding yourself into believing, that reality does not exist independent of your perception of it is no more than a delusion of grandeur. Reality may suck, but reality it still is.][And you know that how? -- Scott]

Reality is what you can get away with. If you can't get away with it it isn't real and has to be dealt with. Saying it isn't real doesn't change that, so the proper course of action is to accept what you perceive as real and deal with it accordingly.

Mar 29, 2013

My 52 yr old sister changes our mutual childhood memories regularly to benefit and support her delusional behavior to rationalize her current abusive behavior. Or maybe it is me that has a different memory of mutually experienced events. Or maybe we're both wrong...point being that the human brain is inherently imperfect and we use memory to adapt to current situations and when it is not a life or death adaptation, our brain plays games with us. I'll take door #2.