Prove the negative of the Monty Hall problem

[imatfaal]

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

 

Monty cannot reveal a door chosen at random - he might reveal the car

[TakenItSeriously]

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

 

Monty cannot reveal a door chosen at random - he might reveal the car

 

Thread is drifting out of context.

 

What makes the incorrect 50:50 answer intuitive is how our intuition reads the MHP. The broken link in this case is in how the reveal is interpreted.

 

Reveals are interpreted by our intuition as a dealer dealing a card. However we also know that Monty always reveals a goat. We have no experiences with dealers who always deal the same card and intuition is only dependent on experiences. Since our intuition sees Monty always revealing a goat as a null experience, we are only left with interpreting it to the experience of dealing a random card, or monty opening a random door.

 

I'm not saying Monty actually reveals a random door, your intuition is interpreting it that way if the 50:50 solution seems intuitively correct to you.

 

To test this, use three cards to simulate the problem.

For example:

 

Usr Ax,2x,2x dealt face down.

You must take the role of Monty and note the position of the Ace.

The contestant can just be a player who always picks door 1 and always sticks.

 

Run the game for a few orbits and the 50:50 answer will no longer seem intuitively correct.

 

Why this happens is that you now have relevant experiences with what happens when Monty always reveals a goat.

Edited July 14, 2016 by TakenItSeriously

[Strange]

 

 

Thread is drifting out of context.

 

What makes the incorrect 50:50 answer intuitive is how our intuition reads the MHP. The broken link in this case is in how the reveal is interpreted.

 

 

I think it is much more likely that people just ignore the door being opened. They see a choice of two (count them: two) doors and hence assume a 50:50 chance. Certainly that is the view of everyone I have talked to about this.

[Daecon]

Think of it this way:

 

I have two cards, an Ace and a 2. I shuffle them and lay them face down, and ask you to choose which one is the Ace. You have a 50:50 chance of guessing correctly, right?

 

Now, after you've made your decision, I deal eight more cards face down next to the first two. Does your chance of having already guessed the Ace go down to 1:10 or does it stay as 50:50?

 

Likewise, if I first shuffle and deal ten cards, and then ask you to guess the Ace, your chance of guessing correctly starts at 1:10. I then remove eight of the non-Ace cards. Your chance of having already guessed the Ace don't increase to 50:50 just because the situation has changed, it stays the way it was when you first made your decision, at 1:10.

 

It's the same principle with three doors instead of ten cards, your original chance was 1:3, so it stays that way even if one of the doors is opened.

 

People disregard that and just see the two doors, and so think "Oh, it must be 50:50 now."

[TakenItSeriously]

I think it is much more likely that people just ignore the door being opened. They see a choice of two (count them: two) doors and hence assume a 50:50 chance. Certainly that is the view of everyone I have talked to about this.

That doesn't explain why people who fully understand why the trade is 1/3:2/3 still see the 50:50 trade as intuitively correct.

[Daecon]

Yes it does.

 

Why do you think it doesn't?

[TakenItSeriously]

Yes it does.

 

Why do you think it doesn't?

No it doesn't.

 

Why do you think it does?

[Daecon]

It's already been explained a few times now why people "intuitively" think the chance is 50:50 when it isn't. If you have a different perspective then why don't you present your viewpoint instead of asking us to guess, or dismissing our other responses?

[TakenItSeriously]

It's already been explained a few times now why people "intuitively" think the chance is 50:50 when it isn't. If you have a different perspective then why don't you present your viewpoint instead of asking us to guess, or dismissing our other responses?

Read post #23

[CharonY]

That doesn't explain why people who fully understand why the trade is 1/3:2/3 still see the 50:50 trade as intuitively correct.

 

Your post only explains why you find it intuitive using assumptions that, judging by the posts, others are not making.

This is not usually how riddles or brain teasers work, as they generally rely specifically on the premise given, rather than trying to figure out what is going on in your mind.

Edited July 16, 2016 by CharonY

[TakenItSeriously]

Your post only explains why you find it intuitive using assumptions that, judging by the posts, others are not making.

This is not usually how riddles or brain teasers work, as they generally rely specifically on the premise given, rather than trying to figure out what is going on in your mind.

Given that the vast majority of the population share the same incorrect illusion, I would hardly think its something unique to my mind and finding a solution would be of particular value for being able to judge the validity of future axioms.

 

Also, I dont think anyone who actually read the solution disputes its correctness. Never the less I gave a method for proving the solution to yourself which would take deck of cards and about 20 seconds to verify.

Edited July 16, 2016 by TakenItSeriously

[Owen Martin]

Hello.  I hope this thread is still trafficked.  I think I solved the problem (been losing sleep).  I believe in what is being called the INTUITIVE answer.  Math is an abstraction, so without the proper framework it doesn't work correctly (aka "word problems"). 

I keep seeing this recurring theme: Car/Goat/Goat     Goat/Car/Goat     Goat/Goat/Car

When in fact the possibilities are: Car/Goat #1/Goat #2     Goat#1/Car/Goat#2     Goat #1/Goat#2/Car     Goat#2/Goat#1/Car     Car/Goat#2/Goat#1     Goat#2/Car/Goat #1     

Then if you are forced to switch the possible outcomes are: 1.  Pick car, shown g1, switch to g2=LOSE     2.  Pick car, shown g2, switch to g1=LOSE     3.  Pick g1, shown g2, switch to car=WIN     4.  Pick g2, shown g1, switch to car=WIN

2 wins and 2 losses.  50/50 if you switch.  

I created an account just for this.  I hope someone reads this...Thanks.

[Ghideon]

Hello @Owen Martin

1 hour ago, Owen Martin said:

I believe in what is being called the INTUITIVE answer. 

Ok!

Your suggestions does not seem intuitive to me, even without using math i get:
-If the goats were identical and not possible to tell apart, would that change the possibility to win a car? No
-If instead of goats there are nothing behind the two doors so that an example is Car/Empty/Empty, would that change the possibility to win a car? No

1 hour ago, Owen Martin said:

Then if you are forced to switch the possible outcomes are: 1.  Pick car, shown g1, switch to g2=LOSE     2.  Pick car, shown g2, switch to g1=LOSE     3.  Pick g1, shown g2, switch to car=WIN     4.  Pick g2, shown g1, switch to car=WIN

2 wins and 2 losses.  50/50 if you switch.  

The 4 possible outcomes does not have the same probability. It seems you think it is a 50/50 chase of picking the car in the first selection. That does not seem intuitive since there is one car and two goats. I prefer to use math to show that 50/50 is incorrect. But I think it is can be  fun exercise to see that by intuition a 50/50 answer seems incorrect. Try this thought experiment: 

First change the rules as a comparison. Let Monty do the first move and open a door showing one of the goats.  Monty must pick a goat, he is not going to reveal the car. Your chance is now 50/50 to get the car from behind one of the two remaining doors.

Compare to the original rule is that you must choose first. Monty must now show a goat, he is not going to reveal the car. Now is it intuitive that the outcome is not 50/50 in this case? Intuition may not give the correct answer, but it tells that 50/50 seems wrong since the rules are changed from the case above where the outcome was indeed 50/50. It does not seem intuitive that you have the same chance as in the case where Monty had to remove one of the goats first.

Logic and "intuition" can help in the analyse but to get the correct answer I prefer math.   

Edited October 13, 2018 by Ghideon
grammar

[Owen Martin]

8 hours ago, Ghideon said:

The 4 possible outcomes does not have the same probability. It seems you think it is a 50/50 chase of picking the car in the first selection. That does not seem intuitive since there is one car and two goats. I prefer to use math to show that 50/50 is incorrect. But I think it is can be  fun exercise to see that by intuition a 50/50 answer seems incorrect. Try this thought experiment: 

First change the rules as a comparison. Let Monty do the first move and open a door showing one of the goats.  Monty must pick a goat, he is not going to reveal the car. Your chance is now 50/50 to get the car from behind one of the two remaining doors.

Compare to the original rule is that you must choose first. Monty must now show a goat, he is not going to reveal the car. Now is it intuitive that the outcome is not 50/50 in this case? Intuition may not give the correct answer, but it tells that 50/50 seems wrong since the rules are changed from the case above where the outcome was indeed 50/50. It does not seem intuitive that you have the same chance as in the case where Monty had to remove one of the goats first.

Logic and "intuition" can help in the analyse but to get the correct answer I prefer math.   

Thanks for the reply! This has been gnawing at me.  Couldn't sleep very well last night.  

However, I did get a few hours and I think my subconscious did some leg work for me.  I don't prefer math.  I think it's an incredibly powerful tool, but without the right logic it's like firing a gun without the right direction.  Also, I won't allow a theory into my worldview unless I can understand it on a fundamental level.  I think it would be akin to simply taking it on someone's word, and I won't do that with deep philosophical and rational thoughts.  

But now that I've slept! I've begun to see a path to reason this out for myself.  And you helped a bit as well.  I had earlier said:

9 hours ago, Owen Martin said:

When in fact the possibilities are: Car/Goat #1/Goat #2     Goat#1/Car/Goat#2     Goat #1/Goat#2/Car     Goat#2/Goat#1/Car     Car/Goat#2/Goat#1     Goat#2/Car/Goat #1     

Then if you are forced to switch the possible outcomes are: 1.  Pick car, shown g1, switch to g2=LOSE     2.  Pick car, shown g2, switch to g1=LOSE     3.  Pick g1, shown g2, switch to car=WIN     4.  Pick g2, shown g1, switch to car=WIN

Now I'm seeing a different way to interpret what I wrote above.  There's a sort of redundancy there.  The first two possibilities are Pick a car, shown either goat, switch to goat and LOSE.  The possibilities that these two things can happen are equal BEFORE either goat is shown.  But time and the unique conditions do a weird trick.  As everyone is saying, there's a 2/3 chance of winning if you SWITCH.  

You can't pick the car twice with one goat shown anymore! It becomes irrelevant whether the host showed g1 or g2.  The possibilities collapsed in such a strange and UNINTUITIVE manner! It's almost like the reason I couldn't figure it out was that I was stuck in the past! I am grateful to "TakenItSeriously" for starting this thread.  I don't actually care about who is correct, the very question of why it seems so obvious at first is confounding! 

The psychological implications of this have got to be profound.  As I was trying to transition to the new paradigm in "opened door world", my brain was screaming halt!  I hadn't resolved the issue yet.  It would've been 50/50 with 2 doors before, but now it's not.  The host hasn't moved anything, we should still be on the same course.  Immediately my brain agreed, it was time to put away the question and move forward.  Don't let those devious Switchmen slow you down! 

HOWEVER, my elder brother was convinced otherwise and I had spent the hours previous arguing without convicting him.  He's fairly open minded in the philosophical realm, so this meant something.  

I walked away from the phone call, immediately discussed it with a close friend at hand.  We reassured eachother.  "These people were confusing the issue!" " Most of these people haven't reasoned it out themselves, they're just a bunch of parrots adopting the supposed intellect of others!" 

But there was something deep within me unsettled.  I read the thoughts of others, tired of the antagonism of forums, and eventually posted my own thoughts CONVINCED that I was right and walked away.  But it wasn't enough.  I shivered with nervous energy.  I laid awake until 5am without thinking of anything in particular.  

Then I woke.  Suddenly I could think differently.

Before, the noise from my mind was so loud I had trouble even moving forward in thought.  Now I woke with a sudden suspicion that I had been wrong.  And here I am.  

WHAT AN EXPERIENCE! I've crossed so many psychological mechanisms to get here! Although I'm still not entirely convinced, with enough empirical data I could be convinced now (that may not have been the case before), and I'm leaning Switchmen.  Before I would question the source, it's methods and intentions, ad infinitum.  

What happened in my mind? Is this a matter of right brain/left brain hoopla? Was the right brain doggedly focused on expediently executing judgement based on my own known theories, while the left brain waited for it to loosen up and chime in? What is the nature of conscious/subconscious? How is it that I can only be convinced of something asleep?! Also, it's quite fascinating how the probability DOES seem to change.  This might have more implications in physical reality than I realize. 

I'm sure my peers think to themselves that I would have been the more intelligent had I grasped the concept sooner-that I was stubborn in my thinking.  That I should've been convinced by the empirical data alone! BAH! Studies lie! I'm a chemist who's seen group bias and the manipulation of data! Hell, I was a highschooler who'd seen those things as well! There's value in conservatism! If you can't understand a theory, don't bed it! Although ironically it was sleep that opened me to the idea...Nevertheless! Skepticism is healthy! 

Perhaps you'd had won and I lost.  Perhaps you're driving that brand new Mitsubishi Mirage right now! But I'll take my goat.  I love and understand goat.  And I'll walk away a rich man in my experience.  I'm still thinking about it...

 

Edited October 13, 2018 by Owen Martin

[JeffJo]

On 7/8/2016 at 1:22 PM, TakenItSeriously said:

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

Those who intuit that the answer is 50:50 are using what is known as the Principle of Indifference on the two closed doors. And it doesn't apply to them. But they shouldn't be criticized too much - the reasons for why it doesn't apply have nothing to do with mathematics, and many who should know better get it wrong in other problems.

If the paths needed to reach two different outcomes are identical, except for the names we apply to the details, then the PoI says that they should have the same probability. That is, if we have no reason to claim one result is more, or less, likely, then we must consider them to be equally likely. If a process has N different outcomes that all have this property, then each has probability 1/N.

This principle is so ubiquitous that we may not realize we are applying it. It is why we say that the car has a 1/3 chance to start behind each of the three doors. To our knowledge, there is no reason why there is more, or less, reason for the car to be behind door #1, #2, or #3.

Let's say that the contestant originally picked door #1, and the host opened door #3. These are just the names that make no difference to the PoI, and using names makes it easier for me to talk about the solutions. Then:

• To say the answer is 50:50, you first have to say that you know of nothing that distinguishes door #1 from door #2.
  • This is definitely not true: it was possible for door #2 to have been opened, but impossible for door #1 since the host would never open the contestant's door.
• Once you have identified a difference, you have to have a reason to suspect that this difference affects the probability of reaching the known result, that door #3 was opened.
  • If the car is behind door #2, a 1/3 chance, then the host was forced to open door #3.
  • If the car is behind door #1, then the host had a choice. Either door #2 or door #3 could have been opened.
    • The PoI no longer applies, and the answer cannot be 50:50.
  • The PoI does apply to which door the host chooses to open when the car is behind door #1.
    • With two choices, the probability of door #3 being opened is halk of what it was before this information.
  • This makes the probability that the car is behind #2 (the first case in this list) twice that of #1.
    • The answer is 1/3:2/3

The reason this is hard to see, is that it was not made clear that we know which door was opened.

+++++

It is the way that the MHP problem is presented that makes it seem self-evident that door #1 and door #2 are equivalent, when they are not. There are other problems where the exact same reasoning should apply, but the presentation isn't the same.

• Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?

The most naive solution here is that "the other" child - the one who may or may not be a boy - must have a 50% chance to be a boy. This is wrong because no specific child was identified as "the boy." That is, if there are two boys, which is "the other"? (This might be easier to see if the family had five children, with at least two boys. Which are "the other three"?) So the error is in determining the set of cases, not applying probability theory to them.

Many experts will explain that in a two-child family, there are three equally-likely (by an idealistic application of the PoI) combinations. Based on birth order, those that that fit the given information are: BB, BG, and GB. If the PoI applies to them, each should have a 1/3 probability, and the answer is 1/3. But if you examine this logic, you can see that it is the same logic used to get 50:50 in the MHP. Only the number of possibilities has changed. The issue is whether the BB possibility is equivalent, under the PoI, to the BG and GB possibilities.

Martin Gardner, who first posed this problem (1959 in his Scientific American column), originally answered "1/3." But he retracted that answer, claiming the question was ambiguous. Specifically, it did not distinguish between the many ways you could know that at least one child is a boy:

• You asked Mr. Smith "Do you have at least one boy?", and he said "yes."
• You met a son.
• Mr. Smith told a story about a son.
• Mr. Smith is a Boy Scout leader.
• etc.

The PoI applies only to the first case. In almost any other reasonable way to learn "at least one is a boy," the PoI does not apply since you could also have learned that there was at least one girl. This has the same result as the host choosing between two doors, and it makes (as Gardner admitted) the answer 1/2.