# Prove the negative of the Monty Hall problem

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I assume everyone here is familiar with the Monty Hall problem.

I'm not looking for the solution.

I'm not looking for another proof for the solution.

I'm not even looking for a full game solution which might incidentally show how the 50:50 solution couldn't be correct..

The challenge is to directly figure out why the 50:50 answer is wrong but more importantly why it seems so axiomatically correct.

I've had the solution for some time, but wasn't sure how to present it. Since it's become a classic logic riddle, I realized this was probably the best forum. At least I hope so, Im still pretty new here.

I'll give it some time and provide a new hint every now and then, before giving the solution.

Hint:

If you find the 50:50 answer axiomatic, than you will flip over the solution.

Hint:

The answer about why we perceive the 50:50 solution as correct will tell us something about how the subconscious perceives reality.

Hint:

Don't think of this as a logic problem,its more like a human nature problem.

PS, I'm posting from an iPad and don't have an editor toolbox. I'd appreciate it if someone could post a hidden text box or spoiler alert box so I can clip the proper formatting elements. Thanks.

Edited by TakenItSeriously
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It isn't paradoxical the answer is that once the host opens the first door with a goat behind it you have a 50:50 chance of getting the car.

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It isn't paradoxical the answer is that once the host opens the first door with a goat behind it you have a 50:50 chance of getting the car.

I was afraid this would happen, Im not asking for the solution or proof.

The challenge is to figure out why the incorrect intuitive answer is intuitive.

Ive cleaned up the op to hopefully clarify this point.

Sorry, it was a mess. My editor is acting up again and it starts becoming a chore to edit posts properly.

Edited by TakenItSeriously
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The challenge is to figure out why the incorrect intuitive answer is intuitive.

It isn't it is just a load of crackpots trolling stupid people.

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It isn't paradoxical the answer is that once the host opens the first door with a goat behind it you have a 50:50 chance of getting the car.

That is quite possibly the worse answer to Monty Hall I have seen. If you simply had a 50:50 chance then why swap - the clever part of the problem is that Monty always reveals a goat and by doing so changes the information you have and thus increases your chances (you have a 1/3 stay and 2/3 switch win probality

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I am still not entirely sure what your question is, but the 50:50 appears to be intuitively correct (whilst being wrong) as people generally dismiss the additional information provided by revealing one wrong door.

Edit: crossposted

Edited by CharonY
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..

The challenge is to figure out why the incorrect intuitive answer is intuitive.

...

Because humans are naturally bad at probability - it is something that a huge majority must learn rather than understand intuitively. There are many things like that - many of the topics here are also unintuitive BUT they do not crop up so regularly in common world experiences.

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It isn't paradoxical the answer is that once the host opens the first door with a goat behind it you have a 50:50 chance of getting the car.

That's the incorrect answer. You can do it just marking out the possibility space with pictures of doors, cars, and goats, or you can do it the Bayesian way and you get the same answer. That answer is not 50/50.

You can run actual trials (as Mythbusters did) and confirm. 50/50 is an incorrect answer.

Horrible quality video to follow:

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That is quite possibly the worse answer to Monty Hall I have seen. If you simply had a 50:50 chance then why swap - the clever part of the problem is that Monty always reveals a goat and by doing so changes the information you have and thus increases your chances (you have a 1/3 stay and 2/3 switch win probality

Correct.

But why is the 50:50 answer wrong and why does it seem intuitively correct.

No proofs for the 1/3:2/3 answer please. Only direct proofs for why the 50:50 answer is wrong.

Because humans are naturally bad at probability - it is something that a huge majority must learn rather than understand intuitively. There are many things like that - many of the topics here are also unintuitive BUT they do not crop up so regularly in common world experiences.

Actually, even prominent mathematicians were fooled by this problem.

Studies have shown that for those who initially got the answer wrong it seemed to be independent of intelligence or education.

That's the incorrect answer. You can do it just marking out the possibility space with pictures of doors, cars, and goats, or you can do it the Bayesian way and you get the same answer. That answer is not 50/50.You can run actual trials (as Mythbusters did) and confirm. 50/50 is an incorrect answer.Horrible quality video to follow:https://www.youtube.com/watch?v=FAljAvR3L4s

Thats, all true but it doesn't address the 50:50 answer directly or why most people think 50:50 is correct.

Trust me the answer tells us important information about how the subconscious perceives reality.

I am still not entirely sure what your question is, but the 50:50 appears to be intuitively correct (whilst being wrong) as people generally dismiss the additional information provided by revealing one wrong door.

Edit: crossposted

Actually, they didn't dismiss the additional information, since they thought their odds changed from 1/3 to 50:50.

So they were wrong despite changing their odds to what they thought they knew was correct.

Your definitely on the right track though.

Edited by TakenItSeriously
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Correct.

But why is the 50:50 answer wrong and why does it seem intuitively correct.

No proofs for the 1/3:2/3 answer please. Only direct proofs for why the 50:50 answer is wrong.

Actually, even prominent mathematicians were fooled by this problem.

Studies have shown that for those who initially got the answer wrong it seemed to be independent of intelligence or education.

Thats, all true but it doesn't address the 50:50 answer directly or why most people think 50:50 is correct.

Trust me the answer tells us important information about how the subconscious perceives reality.

Actually, they didn't dismiss the additional information, since they thought their odds changed from 1/3 to 50:50.

So they were wrong despite changing their odds to what they thought they knew was correct.

Your definitely on the right track though.

It is wrong because it is not right - there would be an internal contradiction if it were to be correct; ie the maths says the probability is 1/3:2/3 (stay:switch) which is not the same as 1/2:1/2

Why people think 50:50 (stay:switch) is correct is because they guess rather than do the maths. If you do the maths you get it right, if you use walking-ape logic you get it wrong. I think - if you really want to investigate the flawed reasoning you have to get into psych rather than maths and that your best avenue would be reading the literally hundreds (perhaps thousands) of internet threads on this very problem. There are lots of misguided souls out there who continue to bang on without the faintest glimpse of enlightenment.

You say "trust me" - with respect No. This is SFN - you should pony up proof that this is manifestation of a deep set internal make up. And where are the studies - must have been done a long time ago as Monty Hall is very very famous and that would completely screw the validity of any test

But there is no mathematical reason why people are wrong - just a lack of mathematical reasoning

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It is wrong because it is not right - there would be an internal contradiction if it were to be correct; ie the maths says the probability is 1/3:2/3 (stay:switch) which is not the same as 1/2:1/2

Why people think 50:50 (stay:switch) is correct is because they guess rather than do the maths. If you do the maths you get it right, if you use walking-ape logic you get it wrong. I think - if you really want to investigate the flawed reasoning you have to get into psych rather than maths and that your best avenue would be reading the literally hundreds (perhaps thousands) of internet threads on this very problem. There are lots of misguided souls out there who continue to bang on without the faintest glimpse of enlightenment.

You say "trust me" - with respect No. This is SFN - you should pony up proof that this is manifestation of a deep set internal make up. And where are the studies - must have been done a long time ago as Monty Hall is very very famous and that would completely screw the validity of any test

But there is no mathematical reason why people are wrong - just a lack of mathematical reasoning

You are correct, in that the reason the overwhelming majority of people thought 50:50 was correct is based upon human nature. But, that is the point of the question. Why does the 50:50 choice seem so intuitively correct?

It is a varidian paradox. Or a result that differs from what seems to be obvious.

I put that in one of the hints, after you had read the OP. Unfortunately, I couldn't think of a way to notify everyone as to when hints would get updated.

I generally update hints when peoples responses in the thread were touching on correct points that were on the right track. So look for more hint updates after this post.

You are also correct in that math is not the proper tool for finding the source of an incorrect solution. At least, not for a comprehensive proof (beyond a reasonable doubt). It's the tool for finding correct solutions.

When looking for the source of an incorrect response, you need to start at that response and work backwards through induction, at least initially. But that is not how most logic problems are approached. Only real world problems are approached that way.

For example, when solving a crime, it is always an induction process at the beginning to follow the clues that will produce all of the facts. At that point you can reassess the problem from the perspective of the past moving forward through a deduction process.

I had intended the following to be a separate post, unfortunately the forum structure prevents that from happening.

Hint update:

I assume everyone here is familiar with the Monty Hall problem.

I'm not looking for the solution.

I'm not looking for another proof for the solution.

I'm not even looking for a full game solution which might incidentally show how the 50:50 solution couldn't be correct.

The challenge is to directly figure out why the 50:50 answer is wrong but more importantly why it seems so axiomatically correct.

I've had the solution for some time, but wasn't sure how to present it. Since it's become a classic logic riddle, I realized this was probably the best forum. At least I hope so, Im still pretty new here.

I'll give it some time and provide a new hint every now and then, before giving the solution.

Hint:

If you find the 50:50 answer axiomatic, than you will flip over the solution.

Hint:

The answer about why we perceive the 50:50 solution as correct will tell us something about how the subconscious perceives reality.

Hint:

Don't think of this as a logic problem,its more like a human nature problem.

Hint:

See post #11 for the approach one would need to take to start trying to solve this problem.

It is not an easy solution. I'm pretty sure mine was the first attempt to prove it. But a 100% valid proof is not possible with regards to human nature. It can only be intrinsically proven to the point of making sense.

Edited by TakenItSeriously
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Because they dismiss the additional info that the host reveals a wrong door and instead and instead reduce the question to a two door problem. I.e. it is assumed that the reveal simply removes it out of the equation, whereas as others pointed out it is still very much part of it.

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Hint update:

Warning, there is no easy method (yet) to solve this problem. Therefore, I will leave it open a little longer than normal and post more hints then your probably accustomed to.

Hint 1

If you find the 50:50 answer to be intuitive, you will probably flip when you discover the answer.

Hint 2

Finding the answer as to why we perceive the 50:50 as being axiomatically true will tell us something about our perception of reality. Which is pretty relevant in the Physics forums.

Hint 3

Don't think of this as a math problem. Math is a tool for finding correct answers, not for understanding incorrect answers.

Instead you should think of it as it relates to human nature. Use induction and deduction to try and solve it.

Hint 4

Human reality is based first and foremost through our first hand experiences from our own five senses.

Human intuition is based upon repeated patterns of experience.

What human experience is the monty hall riddle representing that is a very common experience to the vast majority of the population?

Here's a free hint, don't start with all of human experience when finding the subset that is common to both the riddle and the population.

Edited by TakenItSeriously
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Because they dismiss the additional info that the host reveals a wrong door and instead and instead reduce the question to a two door problem. I.e. it is assumed that the reveal simply removes it out of the equation, whereas as others pointed out it is still very much part of it.

Close, but not quite correct IMHO.

I think the key to understanding this is that our human intuition is not linked to our ability to reason.

Intuition is nothing more than learning through pattern recognition. It was around long before reason or even consciousness existed so it simply does not rely on reason and in fact it works despite our using reason. That is not to say that we have no control over intuition, clearly we can choose to ignore it which many people have finally succumbed to the sheer brute-force of it where too many people shared the opposing opinion and they may have conceded the point but they still would have felt the opposite. However, there are far easier ways to dispel false intuition which I promise to explain after the reveal.

Don't worry its not soft science, it's simply correcting a misperception. But the only way to do that is to be able to identify the source of the misperception. Which is the point of this exercise.

BTW, I've recently updated the hints in the post above if anyone is interested.

Edited by TakenItSeriously
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Hint update:

Warning, there is no easy method (yet) to solve this problem. Therefore, I will leave it open a little longer than normal and post more hints then your probably accustomed to.

Hint 1

If you find the 50:50 answer to be intuitive, you will probably flip when you discover the answer.

Hint 2

Finding the answer as to why we perceive the 50:50 as being axiomatically true will tell us something about our perception of reality. Which is pretty relevant in the Physics forums.

Hint 3

Don't think of this as a math problem. Math is a tool for finding correct answers, not for understanding incorrect answers.

Instead you should think of it as it relates to human nature. Use induction and deduction to try and solve it.

Hint 4

Human reality is based first and foremost through our first hand experiences from our own five senses.

Human intuition is based upon repeated patterns of experience.

What human experience is the monty hall riddle representing that is a very common experience to the vast majority of the population?

Here's a free hint, don't start with all of human experience when finding the subset that is common to both the riddle and the population.

Hint 5

Human intuition is independant of reason.

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I am not sure why there are spoiler tags as there are only a number of assertions and handwaving, but I cannot see any actual data/hints, references or relevant information.

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So... Do you want us to give you our answer as to why the incorrect answer seems intuitive, or to guess whatever answer you're thinking of?

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So... Do you want us to give you our answer as to why the incorrect answer seems intuitive, or to guess whatever answer you're thinking of?

Well, I'm pretty sure my answer is correct and I've tested it on a limited basis among friends, but I don't have the resources to run a scientific study.

I stumbled on the solution because I happened to have the right combination of interests which made it effectively luck. I believe the solution could identify when a false axiom was incorrectly assumed in the future but to be safe, I was also hoping someone else could discover a more reliable method for solving these types of problems before I gave the final solution and method for proving the hypothesis which might bias an original line of thinking.

I am not sure why there are spoiler tags as there are only a number of assertions and handwaving, but I cannot see any actual data/hints, references or relevant information.

The conflict is between a correct answer arrived at using reason vs a seemingly intuitively correct (but wrong) answer.

Therefore the conflict is a function of how intuition works vs reason.

Spoiler tags are used incase someone finds a potential line to solve the problem at some point without reading all of the hints.

Hint update:

Warning, there is no easy method (yet) to solve this problem. Therefore, I will leave it open a little longer than normal and post more hints then your probably accustomed to.

Hint 1

If you find the 50:50 answer to be intuitive, you will probably flip when you discover the answer

Hint 2

Finding the answer as to why we perceive the 50:50 as being axiomatically true will tell us something about our perception of reality. Which is pretty relevant in the Physics forums.

Hint 3

Don't think of this as a math problem. Math is a tool for finding correct answers, not for understanding incorrect answers.

Instead you should think of it as it relates to human nature. Use induction and deduction to try and solve it

In hind sight, I took the relationship between intuition and reason for granted, which I realize is not a fair assumption to make. So I added hints 3.6 & 3.7 to hopefully clarify hints 4 & 5.

Hint 3.6

The conflict of the Monty Hall Problem is that the seemingly intuitively corect (though wrong) solution is different from the correct solution that was arrived at through conscious reasoning.

Hint 3.7

In Physics or global warming, we know that intuitive assumptions are often wrong, because the assumptions are based on experiences that are invalid due to scale or transitional states.

With the Monty Hall Problem, we are dealing with a time independent problem at the human scale so experience should be relavent. Yet our intuition fails as if the associated experiences don't apply.

Hint 4

Human reality is based first and foremost through our first hand experiences from our own five senses.

Human intuition is based upon repeated patterns of experience.

What human experience is the monty hall riddle representing that is a very common experience to the vast majority of the population?

Here's a free hint, don't start with all of human experience when finding the subset that is common to both the riddle and the population.

Hint 5

Human intuition is independant of reason.

I will post the solution tomorrow evening, if you are working on a solution and would like more time please let me know before that time. Edited by TakenItSeriously
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Finding the answer as to why we perceive the 50:50 as being axiomatically true will tell us something about our perception of reality. Which is pretty relevant in the Physics forums.

It might have some relevance to the philosophy or psychology forums. I can't see any connection to physics.

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It might have some relevance to the philosophy or psychology forums. I can't see any connection to physics.

I was referring to the unintuitive nature of physics outside of the human scale of experience

Edited by TakenItSeriously
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The challenge is to directly figure out why the 50:50 answer is wrong but more importantly why it seems so axiomatically correct.

People assume that because there are two doors there is a 50:50 chance. I'm not sure why you are making this so complicated.

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For a detailed look at the Monty Hall problem, click the Wikipedia link.

https://en.m.wikipedia.org/wiki/Monty_Hall_problem

The Monty Hall Problem

There are three doors, behind one of those doors is a new car, behind the other two are goats eating hey.

A contestant is allowed to pick one of the three doors to will win whatever is behind that door.

Before the contestant can see what they've won, Monty opens one of the other two doors to show a goat which you already knew was going to be the case.

Monty then asks the contestant if he would like to swap what's behind third remaining door for what is behind the contestants door.

Should you:

B) stick with your first choice

C) It doesn't matter (coin-flip)

The Monty Hall problem is an example of a Veridical Paradox, which could be viewed as:

A Correct result from Reason does not agree with a seemingly intuitively obvious result.

The Correct Answer from reason: (1/3:2/3)

If ignoring our intuition and calculate the odds for the car behind each door using math, logic, or game theory, we would find that the correct answer is a 1/3:2/3 trade.

Upon first hearing the problem few would get as far as trying to calculate the odds because the 50:50 trade was already axiomatic and there didnt seem to be a mpneed to look further. The common assumption as to what was duping our sense of intuition is that their are only two doors left and therefore it should be 50:50

Surprisingly, however, even when people know and fully understand that the correct answer is a 1/3:2/3 trade. They still feel like the 50:50 trade is intuitively true.

My intorduction to the Monty Hall Riddle:

In my first experience with the problem, I found my own perspective of the intuitive answer a rather unique one.

When i was in the 9th grade some 40 years ago. The MHP was presented to me as a logic riddle,

I had just enough experience with solving logic riddles to have the beginnings of a standard strategy going into a problem.

I would take mental notes on key variables as I would hear them in the riddle.

I would always look at the least obvious answers first while completely ignoring the other options. solutions. If needed, i would move down to the next least obvious answers while ignoring all others, and so on.

When I heard that Monty deliberately picked a goat based upon knowing where the car was, I didnt recognize the mathematical form but, I did note that it was asymmetricle. Therefore I could see that the odds of the doors should diverge.

Despite that assumption the most obvious answer was still 50:50 so I completely ignored the 50:50 case and solved for the trade to see if it was consistent.

Goat Reveal:

D1. D2. D3

Car Goat Goat

Goat Car Goat

Goat Goat Car

We assume the contestant picks D1

Monty reveals goats in the assymetricle pattern below:

Monty Reveals a goat

D1. D2. D3

Car xxxx Goat

Goat Car xxxx

Goat xxxx Car

D1 = Car, Goat, Goat = 1/3

Dx = Goat, Car, Car = 2/3

Correct trade at odds of 1/3:2/3

So notice how I first assumed that two doors would have diverging odds before half way through the problem. Following through with that assumption and my strategy to ignore the obvious. I arrived at the answer more quickly.

The problem got very interesting when asked about why I ignored the 50:50 trade. I explained it to him while looking at it for the first time and saw a 50:50 trade that appeared to be intuitively true!!!

Instead of an intuitive trade 50:50 trade between two doors which support each other before any other considerations are taken into account. My take on the intuitive 50:50 trade was more surreal.

Before I ever considered the intuitive trade, the two doors were already assumed to be asymmetricle and solved at 1/3:2/3 so the state of the problem before I first looked at the intuitive 50:50 solution was known and complete contradiction to the 50:50 trade. Yet even in complete contradiction to all known facts it still seemed intuitively correct!!!?

So I knew something much deeper was going on with how our intuition must work.

Over the years, I mostly paid the MHP no attention other than to offer up improved intuitive and elegant proofs when it came up now and then. They helped to sell the 1/3:2/3 trade but still did nothing to remove the intuitive illusion which I assumed would never go away.

It wasnt like the magician who revealed his secret, and the intuitive illusion of his trick would never work on us again.

Finally, some three decades later, when didnt have a pen and paper to help me explain the true answer. i decided to just simulate the problem in front of her with 3 cups and a quarter. She flipped almost instantly to the 1/3:2/3 solution.

I finally got the illusion to turn off! But I wasnt sure what that meant yet.

Epiphanic Moment

Finally, forty years after first expeeiencing the odd intuitive 50:50 that went against everything I knew to be correct. I noticed a post about the Monty Hall problem in a poker forum where the OP had asked if the Monty Hall Problem related to poker like it related to BJ in the movie 21.

It turns out it doesnt relate to poker or even to BJ for that matter. Even the movie 21 got it wrong. In fact the MHP doesnt relate to anything in real life. But why would it? No one is ever going to invent a game where we reveal a known constant every time. It would be completely pointless. Much like the game show was. Other that giving away stuff.

This explained how gaining the relavent experience would fix the problem. But what was creating the fale intuitive answer? The clue was to find experience resulted in the 50:50 trade.

For a random reveal, the odds fit just right:

Random reveal

Assume D1 was picked initially

And D2 was eliminated randomly.

D1 D2. D3

Car Goat Goat

Goat Car Goat

Goat Goat Car

Then

D3 = Goat, Goat, Car

D1 = Car, Goat, Goat

D1 = D3 (50:50)

So if the valid experience is missing, the brain substitutes an invalid experience so long as it is a close enough match based upon experience alone. And the image of revealing a random door with a goat behind is not different from revealing a goat on purpose. However it only works once, after a few trials the experience is completely replaced with the correct version.

So the hypothesis seems to make sense and is self consistent, but is it consistent with what we know about intuition. How consistent is the hypothesis?

Human Intuition:

Its widely accepted that Human intuition is based upon the sum of our experiences. It could also be said that human intuition is the ability to recognize patterns in our experiences.

For any given event such as dropping a baseball which falls to the ground, the more experienceswe have with consistent results, the more those experiences represent our take on reality. Speaking only for myslef, this was why the odds seemed so bizarrely surreal to me. I had four years experience with playing poker with friends by the time I experienced the intuitive 50:50 problem and it was a powerful illusion.

We also know that Human intuition predates human reasoning, or conscious thought. Therefore, it is probably safe to assume that intuition is not a conscious process and it does not rely on reasoning.

Experience without the benefit of reason certainly explains the similar yet invalid experienc model and it seems to be consistent with common ideas about intuition.

Can we test the hypothesis?

If the 50:50 answer still seems intuitive to you regardless of what you choose to believe about the true odds. Or what proofs you have seen to deny the 50:50 result. You can prove it to yourself very easily. If you don't see it as being intuitive, you will need to test someone else who did find the 50:50 trade to be intuitive.

All you need to do is provide the missing experience that is missing by simulating the game to yourself.

You can use 3 cards such as: As, 2c, 2s dealt face down to represent the doors and prizes.

If running the experiment on yourself, you can play Monty Hall and assume the player always picks the same card and sticks with it. After surprisingly few hands you will feel your intuitive perception will flipflop from the 50:50 trade to the 1/3:2/3 trade.

Note, that this is by no means a scientificly valid experiment. But it wouldnt be that difficult to create a valid experiment. Only don't expect me to pay for it, I'm just a broke retired engineer.

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For a detailed look at the Monty Hall problem, click the Wikipedia link.

https://en.m.wikipedia.org/wiki/Monty_Hall_problem

I would propose the following hypothesis:

The Monty Hall Problem creates a false intuitive result as a consequence of our lack of not having any actual experiences of always revealing the same thing in a game or similar situation. Therefore a logical chain of failures that results in our intuitive experience changing from that of always revealing a goat to create a 1/3:2/3 trade to the false experience of Monty making random reveals that results in an incorrect 50:50 trade that is as real to us as actually experiencing random reveals.

Amazingly we can correct this common intuitive failure by modeling the events of Monty always picking a goat using a simple analog of three cards playing cards As, 2c, 2s placed face down and simulate 5-7 hands should be more than enough to correct our own perceptions of reality.

The Monty Hall Problem

There are three doors, behind one of those doors is a new car, behind the other two are goats eating hey.

A contestant is allowed to pick one of the three doors to will win whatever is behind that door.

Before the contestant can see what they've won, Monty opens one of the other two doors to show a goat which you already knew was going to be the case.

Monty then asks the contestant if he would like to swap what's behind third remaining door for what is behind the contestants door.

Should you:

B) stick with your first choice

C) It doesn't matter (coin-flip)

The Monty Hall problem is an example of a Veridical Paradox, which could be viewed as:

A Correct result from Reason does not agree with a seemingly intuitively obvious result.

The Correct Answer from reason: (1/3:2/3)

If ignoring our intuition and calculate the odds for the car behind each door using math, logic, or game theory, we would find that the correct answer is a 1/3:2/3 trade.

Upon first hearing the problem few would get as far as trying to calculate the odds because the 50:50 trade was already axiomatic and there didnt seem to be a need to look further. The common assumption as to what was duping our sense of intuition is that their are only two doors left and therefore it should be a 50:50 trade.

Surprisingly, however, even when people know and fully understand that the correct answer is a 1/3:2/3 trade. They still feel like the 50:50 trade is intuitively true.

My intorduction to the Monty Hall Riddle:

In my first experience with the problem, I found my own perspective of the intuitive answer to be a rather unique experience which demonstrated that our intuition isn't linked to the simple 2 doors for a 50:50 trade assumption..

When i was in the 9th grade some 40 years ago. The MHP was presented to me as a logic riddle,

I had just enough experience with solving logic riddles to have the beginnings of a standard strategy going into a problem.

I would take mental notes on key variables as I would hear them in the riddle.

I would always look at the least obvious answers first while completely ignoring the other options. If needed, i would move down to the next least obvious answers while ignoring all others, and so on.

When I heard that Monty deliberately picked a goat based upon knowing where the car was, I didn't recognize the mathematical form. I did note that picking only goats was an asymmetrical relationship between doors. Therefore I could see that the odds of the doors should diverge though I didn't know much beyond that.

Despite that assumption the most obvious answer was still seemed to be 50:50 so I completely ignored the 50:50 case and solved for the trade to see if it was consistent.

Goat Reveal:

D1. D2. D3

Car Goat Goat

Goat Car Goat

Goat Goat Car

We assume the contestant picks D1

Monty reveals goats in the assymetricle pattern below:

Monty Reveals a goat

D1. D2. D3

Car xxxx Goat

Goat Car xxxx

Goat xxxx Car

D1 = Car, Goat, Goat = 1/3

Dx = Goat, Car, Car = 2/3

Correct trade at odds of 1/3:2/3

I first assumed that two doors would have diverging odds before knowing about the intuitive trade. Following through with that assumption and my strategy to ignore the obvious. I arrived at the 1/3:2/3 trade directly having never noticed the 50:50 intuitive trade.

The problem got very interesting when asked about why I ignored the 50:50 trade. I explained it to him while looking at it for the first time and saw a 50:50 trade that appeared to be intuitively true!!!

Instead of an intuitive trade 50:50 trade between two doors which support each other before any other considerations are taken into account. My take on the intuitive 50:50 trade was more surreal.

Before I ever considered the intuitive trade, the two doors were already assumed to be asymmetricle and solved at 1/3:2/3 so the state of the problem before I first looked at the intuitive 50:50 solution was known and complete contradiction to the 50:50 trade. Yet even in complete contradiction to all known facts it still seemed intuitively correct!!!?

So I knew something much deeper was going on with how our intuition must work.

Over the years, I mostly paid the MHP no attention other than to offer up improved intuitive and elegant proofs when it came up now and then. They helped to sell the 1/3:2/3 trade but still did nothing to remove the intuitive illusion which I assumed would never go away.

It wasnt like the magician who revealed his secret, and the intuitive illusion of his trick would never work on us again.

Finally, some three decades later, when didnt have a pen and paper to help me explain the true answer. i decided to just simulate the problem in front of her with 3 cups and a quarter. She flipped almost instantly to the 1/3:2/3 solution.

I finally got the illusion to turn off! But I wasnt sure what that meant yet.

Epiphanic Moment

Finally, forty years after first expeeiencing the odd intuitive 50:50 that went against everything I knew to be correct. I noticed a post about the Monty Hall problem in a poker forum where the OP had asked if the Monty Hall Problem related to poker like it related to BJ in the movie 21.

It turns out it doesnt relate to poker or even to BJ for that matter. Even the movie 21 got it wrong. In fact the MHP doesnt relate to anything in real life. But why would it? No one is ever going to invent a game where we reveal a known constant every time. It would be completely pointless. Much like the game show was. Other that giving away stuff.

This explained how gaining the relavent experience would fix the problem. But what was creating the fale intuitive answer? The clue was to find experience resulted in the 50:50 trade.

For a random reveal, the odds fit just right:

Random reveal

Assume D1 was picked initially

And D2 was eliminated randomly.

D1 D2. D3

Car Goat Goat

Goat Car Goat

Goat Goat Car

D1. D2. D3

Car xxxx Goat

Goat xxxx Goat

Goat xxxx Car

Then

D3 = Goat, Goat, Car

D1 = Car, Goat, Goat

D1 = D3 (50:50)

Is the missing deliberate goat reveal experience logically consistent with results of a random reveal?.

• We have no experience to match with Monty deliberately showing a goat every time.
• Therefore we remove the context of Monty's always showing a goat.
• Without the context of always removing a Goat
• we are left with only an image of revealing a single on purpose
• revealing a goat on purpose matches revealing a random goat.
Therefore the random goat reveal is consistent with results.

Is adding the experience revealing only goats consistent with the results normalizing for the 1/3:2/3 trade?

• We have new experience to match with Monty deliberately showing a goat every time.
• Therefore we keep the context of Monty's always showing a goat.
experience of revealing

With the visceral experience of always removing a Goat added through a quick simulation using cards our intuitive expectations become normalized. (Intuitive answer changes from 50:50 to 1/3:2/3.)

So the hypothesis seems to be logically consistent with results and expectations therefore it is self consistent.

But is it consistent with what we know about intuition:

Human Intuition:

Its widely accepted that Human intuition is based upon the sum of our experiences. It could also be said that human intuition is the ability to recognize patterns in our experiences.

For any given event such as dropping a baseball which falls to the ground, the more experienceswe have with consistent results, the more those experiences represent our take on reality. Speaking only for myslef, this was why the odds seemed so bizarrely surreal to me. I had four years experience with playing poker with friends by the time I experienced the intuitive 50:50 problem and it was a powerful illusion.

We also know that Human intuition predates human reasoning, or conscious thought. Therefore, it is probably safe to assume that intuition is not a conscious process and it does not rely on reasoning.

Experience without the benefit of reason certainly explains the similar yet invalid experienc model. It seems to be consistent with common ideas about intuition.

Can we test the hypothesis?

If the 50:50 answer still seems intuitive to you regardless of what you choose to believe about the true odds. Or what proofs you have seen to deny the 50:50 result. You can prove it to yourself very easily. If you don't see it as being intuitive, you will need to test someone else who did find the 50:50 trade to be intuitive.

All you need to do is provide the missing experience that is missing by simulating the game to yourself.

You can use 3 cards such as: As, 2c, 2s dealt face down to represent the doors and prizes.

If running the experiment on yourself, you can play Monty Hall and assume the player always picks the same card and sticks with it. After surprisingly few hands you will feel your intuitive perception will flipflop from the 50:50 trade to the 1/3:2/3 trade.

Note, that this is by no means a scientificly valid experiment. But it wouldnt be that difficult to create a valid experiment. Only don't expect me to pay for it, I'm just a broke retired engineer.

Edited by TakenItSeriously
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So, what you are saying is that our experience that a choice of two items normally has a 50:50 chance is what makes most people think that is the answer?

Well, that's a shocker.

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So, what you are saying is that our experience that a choice of two items normally has a 50:50 chance is what makes most people think that is the answer?

Well, that's a shocker.

No the intuitive error is a result of having no experiences with a dealer who always deals the same card.

Without the correct experiences we instead rely on the experiences of dealing random cards as the intuitive model.

If Monty reveals a random door: we end up with a 50:50 trade

Edited by TakenItSeriously

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