The Monty Hall Paradox explained, for once, in English

I've just read the old series of posts on the Monty Hall problem. The explanations--here and everywhere I've looked--consist of equations (albeit simple), which have never been satisfying to me, and I suspect, to many other people, as explanations. What seems itchy about the explanations that are offered is that we know that Monty will reveal a goat, and that we also know that there will be at least one goat for him to reveal. So it seems counterintuitive that something he will ALWAYS do, no matter what, will affect the outcome. Here's how I decided to restate the apparent paradox in English, rather than mathese:

You pick, let's say, Door 1. Monty now reveals Door 2 (or 3), behind which is a goat. The question is:

Did Monty reveal Door 2 (or 3) because he HAD to (you picked the other goat), or because he CHOSE to (you picked the car)? Since you would have picked a goat two out of three times, logically, two out of three times, Monty was FORCED to pick the remaining goat-door, and therefore you should switch. In other words, it is twice as likely that Monty chose to open the door that he did because he was FORCED to, rather than having a choice and happening to choose that one.

It seems to me that this is very similar to the problem of Restricted Choice in bridge.

I've never bought into the 'math', BECAUSE as you point out, Monty doesn't always open a random door. If you switch, how does your 1:3 chance become 2:3? The 2:3 includes and exposed goat!?!? I ALWAYS thought that the option to switch after exposing the goat makes it a 1:2 situation. Until...

It was explained this way: If there were 1,000 doors, and Monty opened 998, you KNOW that the prize is behind either your door or the last door. Do you REALLY think you were lucky enough to have picked correctly when it was a 1:1000 chance? So, even though there are 998 goats exposed, switching makes your chance 999:1000. The same logic applies to the standard three door scenario.

If Monty opened a door at random, then normal 1:2 logic applies. Similarly, in Deal or No Deal, since suitcases are opened at random, normal logic applies.

I don't know Bridge, so I can't comment on that.

He's opening the door because it's a TV show, and that's how the show worked.

Sorry, that sounds mean. I've read your post a couple of times, and I don't quite understand the point. Can you please explain why it matters if we say he "Had" to or "Chose" to. Maybe I'm just a numbers person. Which isn't good because I'm in HR.

What he means by had/chose is this: If you picked a goat, there was only one door left with a goat, so he HAD to open that one. If you chose the car, Monty could open either door.

Bear in mind that on the TV show, the 'doors' were actually curtains or boxes. Only the final deal used doors, and in that deal, two contestants chose doors, so there was no option for Monty to let you change your mind. The thing where Monty gave a choice was when all three curtains/boxes were in play. And it was actually rare for two curtains/boxes to contain a goat/zonk. Sometimes he opened a curtain/box to show an intermediate level prize before giving the option to switch.

And he didn't always give the option to switch.



That said, in what has become known as "The Monty Hall Paradox", three doors are used, one has a car, two contain goats, and after picking, you are ALWAYS shown a goat behind one of the other doors, and offered the option to switch.

For more info: http://en.wikipedia.org/wiki/Monty_Hall_problem

To fully comprehend it, I had to see the Wizard's example. In particular the example that follows "if the player had a strategy of switching."

https://wizardofodds.com/ask-the-wizard/122

The problem I've had with the Monty Hall paradox is the assumption that I want a car instead of a goat. Quite frankly, a goat makes a pretty good pet.

Quote: konceptum

The problem I've had with the Monty Hall paradox is the assumption that I want a car instead of a goat. Quite frankly, a goat makes a pretty good pet.

Better gas mileage and produces meat and milk rather than simply consuming resources.

Quote: konceptum

The problem I've had with the Monty Hall paradox is the assumption that I want a car instead of a goat. Quite frankly, a goat makes a pretty good pet.

I've had some cars that I would have preferred a goat to, all American cars made in the low-quality 80's, when Ford, GM, etc. knew that there were so many people that would "BUY AMERICAN GOLDURNIT", that they could make their cars out of cardboard and tin and millions of people would still buy them.

And to further what you said, I never got a drop of milk out of my Ford Pinto. And on the flip side, if somebody hits your goat from the rear, the goat doesn't explode in a fireball; it just kicks whoever hit him. Hmmmm. And we all used to laugh at the guy who got the goat as his prize.

I know this thread is old, sorry. Its always a favorite subject of mine. I have read on other boards that the human factor means NOTHING, not sure if thats true or not?

Lets say you chose #2 and #1 is opened by a monkey (lol), not Monty, NO CAR. You are allowed to switch if you want to. Are you suppose to switch to #3? I still dont get it?


Ken

If the monkey chooses randomly (and happens not to show you a car), switching or not doesn't matter. But if the monkey acts like Monty -- that is, it will always show you the goat because it knows where the car is and, under the rules, can't show it to you -- then you should always switch.

Quote: MathExtremist

If the monkey chooses randomly (and happens not to show you a car), switching or not doesn't matter. But if the monkey acts like Monty -- that is, it will always show you the goat because it knows where the car is and, under the rules, can't show it to you -- then you should always switch.

Not saying you're wrong so dont jump on me. I have read on other forums from a few math 'experts' that say you ARE suppose to switch doors. The question was geared around NO HUMAN interaction.


Ken

The way I have read this in the past (not saying I'm right)........human interaction and lets say non-human interaction, are or could be (?) two different answers??

Ken

The species of the host doesn't matter, only the behavior. Wikipedia has a good article on this, and near the bottom is a table that compares different host behaviors. If the host acts randomly (as a monkey is presumably going to do) then switching doesn't matter.

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

Let me explain this problem that is as simple as it gets. Let us assume that there were 100 doors instead of just three. Behind 99 doors are goats, and behind one door is the car. Pick a number between 1-100. I randomly choose the car to be behind door number 25. You choose x. I will open all doors except door 25 and door x. Let us assume that x does not equal 25. Now I will give you the choice of having door x or door 25. Although there are only two doors left it is not a 50/50 proposition. 99% chance it is behind door 25 and only 1 % chance it is behind door x.

The key is that i know where the prize is and only open the doors without the prize. Hope this helps.

Quote: AceCrAAckers

99% chance it is behind door 25 and only 1 % chance it is behind door x.

The key is that i know where the prize is and only open the doors without the prize. Hope this helps.

Most people understand it immediately if you make it 1000 doors.
What they get hung up on is the size of the sample, 3 doors. When
you greatly increase the sample, a child can understand it.

wow, I just noticed the prior unappreciated-genius-in-residence of this website could not get the monty hall paradox!! That does surprise me as whatever else you wanted to think about mkl654321 he was at least fairly smart, and very good at chess.

You're all forgetting three very important points:

1 - Monty Hall knew where the car was hidden.

2 - He did not always open a door and give the option to choose.

3 - He wanted want to avoid giving away the car.

As such, it was very rare for him to give you the option to switch, unless you already had selected the car.

A discussion of this was initiated here:
https://wizardofvegas.com/forum/off-topic/general/4396-the-monty-hall-paradox-interview-with-monty-hall/
It includes a link to an article which includes an interview with Monty Hall himself.

Quote: odiousgambit

wow, I just noticed the prior unappreciated-genius-in-residence of this website could not get the monty hall paradox!! That does surprise me as whatever else you wanted to think about mkl654321 he was at least fairly smart, and very good at chess.

Its called an adult asking a question and then getting feedback. The only thing that DOES surprise me are foul comments and I did mention, I have read contradicting answers from other boards.

Ken

Quote: DJTeddyBear

You're all forgetting three very important points:

1 - Monty Hall knew where the car was hidden.

2 - He did not always open a door and give the option to choose.

3 - He wanted want to avoid giving away the car.

As such, it was very rare for him to give you the option to switch, unless you already had selected the car.

A discussion of this was initiated here:
https://wizardofvegas.com/forum/off-topic/general/4396-the-monty-hall-paradox-interview-with-monty-hall/
It includes a link to an article which includes an interview with Monty Hall himself.

Thank you for the reply. I am mainly interested in the answer with NO human interaction, which I guess I got.

Ken

Quote: mrjjj

Quote: DJTeddyBear

https://wizardofvegas.com/forum/off-topic/general/4396-the-monty-hall-paradox-interview-with-monty-hall/
It includes a link to an article which includes an interview with Monty Hall himself.

Thank you for the reply. I am mainly interested in the answer with NO human interaction, which I guess I got.

That's fine, but remember one thing:

It's difficult / impossible to set up this problem without human interaction.

FYI: In the interview, Monty agrees with the math, but also explains why the math is meaningless.

Not really. You could have an RNG choose a door, or a monkey choose a door, a 3 year old choose a door etc.

Ken

Quote: mrjjj

Not really. You could have an RNG choose a door, or a monkey choose a door, a 3 year old choose a door etc.

The essence on solving Monty Hall problem is the clarification of the hosts behaviour.
Based on his behaviour, the player can apply the appropriate strategy maximizing the probability of selecting the prize.

If the hosts behaviour is to always show a goat, best player strategy is to always switch. You will win 2/3.
If the hosts behaviour is to try to keep his car, he will not offer you a switch when you selected a goat. Then best player strategy is to stay on your choice. You will win 1/3.
If the hosts behaviour is to open a random door including the car (the host is a monkey), every player strategy is identically. You will win 1/3.

If you know nothing about the host (i.e. some alien beams you up, and plays the game), you can choose a strategy of randomly picking a remaining door when the host presents you a goat. This strategy will guarantee you the car by 1/2 if you ever get to see a goat.


The key to best strategy is knowing the hosts behaviour. If the behaviour is not part of the problems description, the problem is ill-defined.