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.

Mar 29, 2013

Concerning Schroedinger's cat, leaving aside what his intentions were or not... Who put the cat in the box in the first place? Surely he knows if it was alive or dead or not - it didn't just appear there!You could give it a little shake. If it's alive, you'll know without having to open it. Believe me, you'll know.

Seems kind of odd to think that the only way you could know if something is alive or dead is by looking at it with human eyeballs when other sense we have could easily answer it for us. Have you SMELLED a dead cat?

The cat had a fixed state long before it entered that box; just because YOU don't know what state it was in doesn't mean SOMEONE doesn't. Like the cat. Or the janitor who didn't want to fill out "dead animal" paperwork.

What if you had a box with an alive or dead cat behind one of three doors and a gameshow host who was clearly a dog person... Would it be on TLC? Or is that more of an A&E channel kind-of show?

This was either very wise, or mindbogglingly ignorant... I think I will choose wise.

Mar 29, 2013

Scott, You made my head hurt then I realized you're my long lost brother. Remember how you were always extrapolating theories from the minutiae of life at the dinner table and mom would always tell you to be quiet and drink your milk? Whew! Glad I remembered, it all makes sense now.Mar 29, 2013

I've had two college-level professors tell me that the whole Schrodinger's cat thing is mainly an example of how great scientists sometimes majorly misunderstand each other. You may ahve made a valid point, but I wouldn't use Schrodinger's cat as evidence of it.
Mar 29, 2013

[I wonder how often the rest of us rewrite our histories.]Constantly. Usually along the lines of "I can assure you officer that I was wearing trousers when I entered this park. Right up until the point I was abducted by aliens. When I awoke I was as you see me now. Have you read any SchrÃ¶dinger lately?"

Mar 29, 2013

It's not hard.When you pick a door you have a 1 in 3 chance of getting it right, 2 in 3 chance you got it wrong.

When the host opens a door and you switch, you move over to the 2 in 3 section.

Mar 29, 2013

There's nothing wrong with the Monty Hall problem. This sort of chance estimation is not a fundamental part of reality (except perhaps in quantum scenarios); it is a tool for coping with ignorance. The chance estimates change because our data changes. However the hard reality is that it either is or is not behind a particular door. It might not feel sensible to you, but there's no reason to except our brains are flawless. It is evidence of jack squat.
Mar 29, 2013

Scott,Congratulations on your comment on Telanis' post. Could not have put it better.

But now after grgeil's post I understand it.

There are two options open to you. a)do not switch or b)switch

A.

If you do not switch, there is a total of 1/3 probability that you will get the car. Monty subsequently opening a door does not change the odds, since you have already chosen one out of three doors.

B.

If you do switch, you will surely win if you started with a goat door. And the odds of starting with a goat door are 2/3.

Therefore B. has double the probability of A.

Thanks grgeil.

Mar 29, 2013

People overthink the Monty Hall Problem with too much math, probability and percentages. It is quite simple...if you picked the car initially (33.33% chance) you will lose the car. If you picked a goat initially (66.67% chance) you are going to get the car. For the percentage lovers: 100% of the time that you initially picked the car you will lose it. 100% of the time that you initially picked the goat you will win the car instead.
Mar 29, 2013

One thing a lot of people don't realize about probability is that probability is about knowledge. Or rather, it's about things that we don't know, using information about things that we do know.For example, in the Monty Hall problem, your odds of picking the door with the car are 1/3. But what if I told you (accurately) which door the car was behind? Now what are your chances of finding the car? Suddenly your odds of finding the car are 1 to 1. If you know where the car is, you can pick the correct door every time.

That's what happens when Monty opens the door. He doesn't open a door at random, otherwise he might find the car. No, he always opens a door with a goat. Monty has knowledge about where the car is, and he uses that knowledge to make his decision as to which door to open. If he didn't, he might open the door with the car! But because Monty used information unbeknownst to us to decide on which door to open, that gives us information about where the car is. Not as good of information as we would like, but enough to throw a wrench into our intuition about probabilities.

Mar 29, 2013

Others here have made the point that Schroedinger's Cat was a thought experiment; it was an attempt to belittle the idea of the Copenhagen interpretation. Schroedinger's Cat ignored the difference between the quantum level and the macro level.A very oversimplified way to look at it is thus: matter is made up of energy in which the quantum states have already coalesced into their 'observed' states. Waves that are still in their quantum states are different than those that have already become observed, or in the macro, are matter.

Unobserved waves may be said to exist only in their different probabilities of being in one state or another, but the cat in the thought experiment is not. It is not existing in a quantum state until it is observed. It's already one or the other; dead or alive. Observation makes no difference in that case.

The Monty Hall problem seems unintuitive only because people tend to ignore the original state of the problem. The result is correct, whether it seems so or not.

I don't believe that having a problem that is challenging to understand somehow means that reality is a state of mind. Think of the inertial reference plane in relativity theory. What you observe depends on your inertial reference plane. But what is happening is real. Reality exists whether you choose to recognize it or not.

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]

Mar 28, 2013

For what it's worth - somewhat related, but independent of math. In "Why Everyone (Else) Is a Hypocrite: Evolution and the Modular Mind" Robert Kurzban discusses some ways the brain (or mind - but I'm not going there with _this_ group {smile}) "lies" to itself (or, in his view, perhaps "its selves").
Mar 28, 2013

I like to view the universe as being in a state of creation, and that creation is a matter of perception. All our memory, all of history, and all of reality for that matter is simply the resolution of what we perceive in this instant.I perceive myself sitting at a computer, typing this. My memory of how I got here is simply a reconcilliation, a story I tell, that explains how I got to this state of perception. If I allow for the existence of others, then that explains why my history must resolve with everyone elses, or else I am crazy.

What we think of as reality is simply everyone agreeing on what we perceive, and is not necessarily contigent on the things leading up to this shared perception.

[I agree. -- Scott]

Mar 28, 2013

Actually, the way you described it (and the way it's described 99% of the time) your "solution" to the Monty Hall problem is wrong. The right answer is "there's not enough information to know if switching is a good strategy or not."You've offered a look at playing the game once. You don't know if it's played the same way every time. For example, maybe Monty doesn't know where the car actually is, so sometimes he actually opens a door revealing the car. Then the choice is easy, but of course this becomes a kind of dumb game. That's the intuitive error that people make--thinking he opened a door "at random." His choice of door was not at all random.

More importantly, his choice TO OPEN A DOOR AT ALL might not be random. You've described him opening a door once, and assumed he opens a door every time. Or, more to the point, you've assumed that whether or not he opens a door and gives you a chance to switch is independent of whether you chose correctly to begin with. I could easily modify the game as such: If you've chosen correctly, he opens one of the other door and asks you if you want to switch. If you've chosen wrong, he simply opens the door you chose and reveals that you lost. Outwardly, there's no way, given the game you've described, to know if Monty has done this. My version of the game would actually be kind of fun, because it would screw over people who think they're good at math.

Of course, I also imagine an Elbonian player who has no use for a car that would just get stuck in the mud, but could totally use a free goat.

As for the actual point of your post...what was the question again?

Mar 28, 2013

The Monty Hall problem is easier if you make the numbers bigger.Imagine you have a stack of 52 playing cards, and I tell you that just 1 of them is the winner. Just 1. So without telling you anything else, you already know that 51 of the cards are *not* the winner. If you take 1 card, and leave me with 51, there is still just 1 winner and 51 losers. My pile, therefore, can have either 51 losers or 50 losers plus the 1 winner. Always. So a minimum of 50 losers. My showing you 50 losers at this point doesn't tell you anything, no new information is gained. But the size of my stack didn't change either. Which would you rather have, the 51 cards or the 1? The 51 of course. Knowing that 50 of them are losers doesn't change that, because 50 of them *always* have to be losers.

Now substitute 3 for 52 above, and 2 for 51, and 1 for 50. There is still just 1 winner.

Mar 28, 2013

I belatedly realized that in my haste to be smugly pedantic, I didn't actually address any of the real content of the post.[ I wonder how often the rest of us rewrite our histories. ]

I think the answer to this is "all day, every day". Memory is not the data bank that it is typically thought to be. Memories fade, change, and appear brand new all the time. Eyewitness testimony is inherently suspect.

Here's an example: I have a very clear, very sharp memory from childhood of my cat talking to me. I can see the backdoor where it happened, the light streaming through the window next to the door. I can see and hear Junior (my childhood cat) turning around and looking at me, asking to be let out. I can feel the shock and surprise of hearing a cat talk.

Obviously, I don't really believe my cat talked to me. It was probably a dream that I had, which in the intervening years has been "promoted" into a real memory. That doesn't change the fact that it FEELS completely real to me, as real as any memory I have from that time.

Mar 28, 2013

I think your two examples are both examples of us intuitively being wrong, but the two examples have little else in common. Monty Hall demonstrates how bad we are with probability and statistics - ask anyone who understands Bayesian statistics. The cat being dead and alive at the same time isn't so much a math issue, as it is a misunderstanding (or non-understanding) of reality. Are there really two cats smeared out, which solidify when open the box, or are there two versions of us, who each see a cat in a different state? People who say they understand reality by understanding quantum mechanics are wrong - quantum mechanics is not compatible with relativity. We're only beginning to understand these things.
Mar 28, 2013

Can we go back to yesterday's post? This one makes my brain hurt.[Yesterday wasn't real. -- Scott]

Mar 28, 2013

A couple of things. First of all, his name is SchrÃ¶dinger, or Schroedinger if you don't like umlauts, not Schrodinger. It's a different letter. You wouldn't want people writing "Scott Adums", would you? Actually, that would be pretty funny.Second, his thought experiment involving the cat wasn't a math problem (except in the sense that everything involving numbers is math), it concerns the quantum unpredictability of matter, i.e. it's physics.

(And with the pedantic side of my brain now quiescent, we come to the real point, which is...)

Third, SchrÃ¶ndinger's thought experiment was actually the OPPOSITE of what most people think it was. He was, for all practical purposes, making fun of the idea of a cat being both alive and dead. The cat story was a demonstration of why that school of thought (known as the Copenhagen Interpretation) was wrong. Ironically, in the years since, people have turned his thought experiment around to mean what he was trying to disprove.

Mar 28, 2013

I need to point out that mathematics itself has inherent limitations on what it can prove.https://en.wikipedia.org/wiki/GÃ¶del's_incompleteness_theorems

Mar 28, 2013

A couple users have said this, but to be more explicit: Schrodinger's Cat was intended to point out how absurd it is to think that a cat could be both alive and dead at the same time. It is not possible. When your brain fails to understand how it could be possible, that is expected, and your brain is correct in thinking it is impossible.The Monty Hall problem just indicates that people are bad at logic, that's all. Nothing new. After you pick one door, view the other two as a group. That group's chances are 2/3, obviously. Opening one of the doors does not affect the group's chances; what is behind the doors did not change at all. Probabilities cannot change if the situation does not change. Thus, the chance of the group concealing the car is still 2/3; the fact that only one door is left to be opened is irrelevant.

[About the cat, the point of the thought experiment was to show that something about our understanding of reality is wrong. Beyond that, we can't really say which part is the wrong part.

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]