Monty Hall Problem

[sol invictus]

Maybe I'm misunderstanding someone's position here, but this is the way this dispute appears to me:
Articulett: chance of winning via sticking with the first choice no matter what happens is just P(first pick is car).
Others: incorporating the added information once an event has happened gives the conditional probability P(first pick is car|info), which is not equal to P(first pick is car)
I don't understand why there is such a case of miscommunication. Those two claims are perfectly consistent.

Those two statements are correct (and therefore consistent). I repeated over and over to articulett that no one was disputing that the odds are 1/3 before taking into account the information from Monty opening the door, but that after taking it into account the odds change. She said I was wrong, and that the odds never change.

It's a bit like arguing with someone that says "the weather is clear" because it was clear on April 4th 1963, and when you point out its raining outside responds "the weather never changes" because, indeed, the weather on Aril 4th 1963 never changes.

 

[articulett]

Can you please get a smart person to confirm or can you cut and paste from the article that says your odds change when you are given the option of switching doors.

If you can't do this... then clarify what you mean. Your odds change from what to what? And you say this happens when he asks you "do you want to switch doors", correct? What does this mean? Does this mean anything to anyone besides Claus? What is it your odds change to? From what? Given what reasoning?

And you think you are an expert on this problem? Claus, has anyone ever given you a reason to feel like you have expertise on this problem? Has anyone other than you actually said your odds change when you are given the option of switching?

If not 50-50-- what odds are changing to what exactly... give me numbers please.

My claim is that your odds never switch. That you always have 1/3 a chance of winning when you don't switch and 2/3 a chance of winning if you do. Always-- when he gives the option and before he gives you the option.

You agree that you have a 1/3 chance of winning if you were never given a choice, right? How does his showing you the goat, change the odds that the car is behind your door-- that is, how does it change the odds that you should keep your first choice-- What do you think the odds change to and what choice would you make with this new information?

I say, it doesn't change the odds of your first choice being a car- it just lets you know that the remaining door is less likely to be a goat. Do you agree? Do you agree that the Monty Hall problem shows that, contrary to intuition, you double your odds by switching (from 1/3 to 2/3)? If not, you are wrong. If so, then it's a communication problem.

In either case, I have not been wrong-- but you derailed this thread to claim that I was --and alluded to your imagined expertise on the topic claiming that it was me who didn't understand something. This is something you do quite frequently. I don't think anyone responds well to such behavior.

 

[sol invictus]

And then other people brought up other scenarios... but they either involved the possibility that a car could be shown (rather than a goat) and/or the possibililty that you wouldn't be allowed to make a choice... so that changes the game... it doesn't change the fact that you have a 1/3 chance of winning the car by staying with your first choice.

Yes, articulett, it does. Look - in any version of the game, if it's played over and over again and you always stick with your first choice, the number of times you win will be approximately 1/3.

No one is disputing that.

However, in the version where Monty does not know where the car is and opens a goat door, your odds of winning by staying CHANGE to 1/2. That statement is not in conflict with the one above - the information provided by Monty allows you to narrow down which type of game you're likely to be in (it tells you you're not in a game where the car is behind door 2, if that's the one Monty opened) and taking that into account CHANGES THE ODDS.

 

[sol invictus]

Can you please get a smart person to confirm or can you cut and paste from the article that says your odds change when you are given the option of switching doors.

I think what CFLarsen is saying - and he is free to correct me if not - is that in the standard problem the odds the car is behind door #3 (if, say, you picked #1 and Monty opened #2) change from 1/3 to 2/3 when Monty opens door #2, and hence your odds of winning by switching to #3 also change from 1/3 to 2/3.

In any case that is a correct statement.

 

[CFLarsen]

Those two statements are correct (and therefore consistent). I repeated over and over to articulett that no one was disputing that the odds are 1/3 before taking into account the information from Monty opening the door, but that after taking it into account the odds change. She said I was wrong, and that the odds never change.

I have no idea why articulett is stuck in the first part of the scenario. Nobody has argued that the chances aren't 1 in 3 of choosing the car at the beginning. Nobody.

What articulett doesn't understand is that the changing of the odds is the whole point of the Monty Hall Problem.

It is easy to determine what the odds are at the beginning. You have three doors, one of them have a car, what are the chances of you choosing the door with the car? 1 in 3. Any kid can do that. But that is just a trite probability problem.

The point of the Monty Hall Problem is that the conditions change, from the time where you have to choose between 3 doors, to the time where you have chosen one, are told what is behind one of the remaining doors - and given the option to switch.

I can understand why her students have such a hard time with this. Not because it is a tough nut to crack, and certainly not because some of them are boys. She is just really, really bad at explaining even a simple setup. Her explanations are way too long and unnecessarily complex. Instead, they are downright confusing.

A lengthy stream-of-consciousness is not a substitute for a clear explanation. Complexity is never preferable to clarity - especially when it comes to logical problems like this one.

 

[CFLarsen]

I think what CFLarsen is saying - and he is free to correct me if not - is that in the standard problem the odds the car is behind door #3 (if, say, you picked #1 and Monty opened #2) change from 1/3 to 2/3 when Monty opens door #2, and hence your odds of winning by switching to #3 also change from 1/3 to 2/3.

In any case that is a correct statement.

Yes, yes, yes. The odds change, because the conditions change.

When teaching logic, it is bad paedagogy to complexify. Instead of teaching students the issue, you confuse them, thereby making them think that they are stupid and you are clever. That's not teaching, that's just trampling on your students.

 

[articulett]

That's what YOU Are doing CLAUS! Recall-- I said this:

A funny thing about humans is that they tend to go with their first choice-- as soon as they place a bet they tend to see their odds as increasing-- they feel more confident with their choice... but your odds don't suddenly change when you place your bet or pick your door... your feeling that you are more likely to win, does not make you more likely to win. This is true even with lottery winners. Before playing, they may correctly assess their odds-- but once they've committed, they "feel" as if their odds go up.

You responded quoting the bold with this challenge:

That's precisely what the "Monty Hall problem" shows: That your odds do change when you pick a certain door.

I was clearly talking about your first choice... Your odds are never better than 1 in 3 that you have chosen the car when he shows you the goat--so your odds can never be better than 1 in 3 by staying with your first choice. You replied that your odds DO change. You either changed the situation when you found out you were wrong and decided to add more pedantry to cover for your anger at my showing you to be wrong-- or you didn't pay attention in the first place in your eagerness to prove me wrong. MY statement above IS correct. Your response that is specifically in regard to that statement is NOT correct... it does not imply the sol invictus scenario that you latched onto to save your butt--but I can see why you would want to claim it does... because then it looks like a language problem rather than the fact that you are being exactly the kind of person you are lecturing others (specifically me) not to be.

If you had an ounce of character you'd admit it and apologize.

 

[CFLarsen]

Where did I say that the odds suddenly became 50-50?

 

[sol invictus]

I was clearly talking about your first choice... Your odds are never better than 1 in 3 that you have chosen the car... not in any case. You replied that your odds DO change.

articulett - he's talking about the standard problem, and the odds you will win if you switch to the door Monty doesn't open. Not the non-standard problem, and not the odds you will win if you stay.

He didn't make that very clear, in part because he is a bit of a troll (judging by his posting history). But he is correct in this case.

 

[articulett]

My students don't have a hard time. I have about 4 boys out of a hundred every year that sound like you however... but the other students and the trials... eventually shut them up. Some still stay mad because they don't like to know that they can be wrong. But most are glad to learn the way their brain fools them.

BTW, your response to Sol makes no sense... so NOW the conditions change, eh?-- Before it was the odds... Neither make sense. That's why you can't find such a quote.

The only thing that changes is that you now know that the door with the goat that he revealed has 100% chance of not being a car-- leaving the remaining door with a 2/3 chance of being a car. That's it.... that's the only thing that changes. Not the odds of your first choice being the winner. Not the odds that you should stay or switch... not the "conditions" as you now claim... just the fact that you now know the car is NOT behind that door.

 

[articulett]

articulett - he's talking about the standard problem, and the odds you will win if you switch to the door Monty doesn't open. Not the non-standard problem, and not the odds you will win if you stay.

He didn't make that very clear, in part because he is a bit of a troll (judging by his posting history). But he is correct in this case.

Correct in what way... that your odds change? What odds change to what exactly. I maintain that your odds are alway 1/3 by keeping it (and I'm talking the standard scenerio) and always 2/3 by changing it. Are you saying something different? If so-- you are wrong.

 

[articulett]

Where did I say that the odds suddenly became 50-50?

Why don't you clarify Claus...

I said that you never have more than a 1/3 chance of being correct by switiching... You claimed the odds change when he offers the choice.... what odds change to what, Claus?

I claim that even before he shows you a goat your odds of winning the car are doubled (in the classic scenario) if you switch-- this doesn't change when he shows you the goat... the only thing that changes is you now are left with the option you must switch to in order to get those 2/3 odds.

So what odds are changing when he reveals the goat-- You claimed that the odds change... and then you claimed that the condition changes... I claim that the only thing that changes-- is that you can no longer choose the door the host revealed-- that doesn't change any odds or any conditions as far as I can see. If that is what you meant-- then you've expressed it so poorly that I doubt you could convey the problem to anyone. I can and have conveyed it to many. I am used to your kind of pettiness and denial-- but usually it's coming from 15 year old boys not grown men.

I have all Freshman this fall (14 year olds). I suspect the majority of them will be able to understand the Monty Hall problem better than you AND communicate it better to others.

 

[CFLarsen]

My students don't have a hard time. I have about 4 boys out of a hundred every year that sound like you however... but the other students and the trials... eventually shut them up. Some still stay mad because they don't like to know that they can be wrong. But most are glad to learn the way their brain fools them.

Ah, OK. Now, it is only a few "boys". Take a look at your earlier blanket statement:

Yes Claus, I read it. I do this demonstration in my class every year, and I'm used to 15 year old boys sounding just like you.
...
I actually love teaching this stuff, because there's always egotistical young men in class who cannot believe that their brains can fool them. And they are the easiest to fool and their egos get in the way of their understanding this. But it's a valuable lesson to learn the ways your brain fools you. And it's fun, because as each student has an "aha" moment they begin to try and teach the blowhards... we can recreate with cups and keep track of the odds--

I'm right. You're wrong. Again. Deal with it.

Why do you single out boys? Do you have any evidence that girls are better at logic than boys?

BTW, your response to Sol makes no sense... so NOW the conditions change, eh?-- Before it was the odds... Neither make sense. That's why you can't find such a quote.

The only thing that changes is that you now know that the door with the goat that he revealed has 100% chance of not being a car-- leaving the remaining door with a 2/3 chance of being a car. That's it.... that's the only thing that changes. Not the odds of your first choice being the winner. Not the odds that you should stay or switch... not the "conditions" as you now claim... just the fact that you now know the car is NOT behind that door.

Where did I say that the odds suddenly became 50-50?

 

[articulett]

Yes, articulett, it does. Look - in any version of the game, if it's played over and over again and you always stick with your first choice, the number of times you win will be approximately 1/3.

No one is disputing that.

However, in the version where Monty does not know where the car is and opens a goat door, your odds of winning by staying CHANGE to 1/2. That statement is not in conflict with the one above - the information provided by Monty allows you to narrow down which type of game you're likely to be in (it tells you you're not in a game where the car is behind door 2, if that's the one Monty opened) and taking that into account CHANGES THE ODDS.

Yes, but that's not the classic story... in Claus' scenario-- Monty always reveals a goat and always gives you a choice.

In the blind scenario-- the host reveals a CAR 1/3 of the time-- so he either takes away your choice or gives you a 100% of a chance of winning by allowing you to choose... so either way... you have a 1/3 of a chance of winning the car by remaining with your first choice (that's average of 0% of 1/3 of the time and 50% of 2/3 of the time).... see? (conversely you have a 2/3 chance of winning by switching... 100% of the 1/3 of times he shows you the car and 50% of the 2/3 of the time he reveals a goat.) You can't really throw out the fact that 1/3 of the time-- he will be revealing a car if he's choosing blindly.

But Claus gets mad if we are talking about different scenarios like that. We are only talking about the scenario where the host always shows a goat and always gives a choice... Claus is very clear that we are ONLY talking about that scenario.

I maintain it's also the same odds when you don't know if this is the case or not... but that's another chapter...

http://www.shodor.org/interactivate/activities/SimpleMontyHall/
that's the simple applet... the door you choose first has the pointer on it...

You can test the more advanced scenarios here:
http://www.shodor.org/interactivate/activities/AdvancedMontyHall/

 

[articulett]

I'm not singling out boys... in my years of teaching... it's the boys who are always angry that their conviction has been proven wrong... not all the boys.... but only the boys. The girls are embarrassed, but they never had the conviction in their rightness the boys have. That's just my experience.... Every year I have 3 or 4 students that are "like you"-- very resistant and angry-- and they are always boys... But when we run the tests.... they can't deny the numbers. But they will be angry at me for exposing the fact that they can be fooled... it's just like the guy in Randi's class when he exposes the astrology trick (in his Nova program)... instead of being humbled by the knowledge, they are mad at the messenger. It's rare... but it's less rare in boys then in girls.

I never made a blanket statement. I've always maintained that it's just a few boys....I'm not responsible for what you add to my words in your head.

You never said 50-50... you said your odds changed... and in my experience most people who believe the odds change, believe that the odds become 50-50 because there are 2 choices.

YOU SAID THE ODDS CHANGE-- SO TELL US-- WHAT ODDS CHANGE TO WHAT? Moreover you said the odds change WHEN he offers you a choice. On top of that you said this in RESPONSE to my statement that your odds are never better than 1/3 for the first choice... it's always 1 in 3 if you keep the first choice.

If you don't answer what you meant when you said the odds change then I will take it as an admission of your error in response to my words. I was right-- you claimed I didn't understand the problem.

I recognize your straw man as an attempt to throw everyone off of the fact that YOU WERE WRONG and you can't admit it.

You make allegations about me all the time because you are angry that I revealed your error. But I'm not claiming that I am the only person you do this to.

 

[sol invictus]

Correct in what way... that your odds change? What odds change to what exactly.

I told you that very clearly in the post you quoted. If you can't be bothered to even skim the posts you respond to, there's no point in continuing this.

 

[CFLarsen]

Yes, but that's not the classic story... in Claus' scenario-- Monty always reveals a goat and always gives you a choice.

It's not my scenario. It's the Monty Hall scenario.

Don't make it look as if I am inventing special conditions, or describing special scenarios. I am describing the Monty Hall Problem.

I'm not singling out boys... in my years of teaching... it's the boys who are always angry that their conviction has been proven wrong... not all the boys.... but only the boys. The girls are embarrassed, but they never had the conviction in their rightness the boys have. That's just my experience.... Every year I have 3 or 4 students that are "like you"-- very resistant and angry-- and they are always boys... But when we run the tests.... they can't deny the numbers.

But "not singling out boys"...

Where did I say that the odds suddenly became 50-50? Are you ever going to provide evidence of that claim?

 

[sol invictus]

I claim that even before he shows you a goat your odds of winning the car are doubled (in the classic scenario) if you switch-- this doesn't change when he shows you the goat...

That doesn't make any sense. Switch to what? If you switch to a random door (your only option, since Monty hasn't yet shown you anything) your odds remain at 1/3.

 

[articulett]

I agree... but the only scenario you want to discuss is the one where he always reveals a goat and he always gives a choice... That IS the classic problem... and it is the only one you want to discuss... You showed anger at me when I discussed others because other people brought up such scenarios. I maintain that my statements are true rather it's the classic scenario or you don't know what scenario you are in. They are true for every case that you would agree with-- plus others.


And I already said that you never claimed 50 -50. It was my presumptions because you said the odds change and that is what everybody else has meant when they say the odds change... they think your odds suddenly become 50-50. You never explained what you meant by "the odds change"-- though you insisted that "the odds changing" is what the Monty Hall problem is about.

So tell us what you meant when you claimed that the the precise thing the Monty Hall problem shows is that the odds change when the host gives you a choice. What the hell did you mean? You said it in response to my bolded statement above. If you didn't mean that the odds suddenly become 50-50-- What DID you mean?? I can think of no way to interpret that statement in response to my statement. You have never clarified. And I think I know why. It's because your odds DON"T change-- my statement was correct and your pedantry was revealed for the sophistry it was.

No one but you thinks I singled out boys... I made it quite clear that in my experiences most students aren't belligerent during this lesson... but the ones that are-- have always been boys. I can't change those facts... and revealing those facts is not singling out boys even if you (singular) imagine it is. Even with the little smiley it doesn't.

 

[articulett]

That doesn't make any sense. Switch to what? If you switch to a random door (your only option, since Monty hasn't yet shown you anything) your odds remain at 1/3.

Yes but he must show a door and the door will be a goat... and so your odds are always doubled from 1/3 (by staying) to 2/3 by switching. You don't have to know what door to switch to... just that you will switch to the door he doesn't open--because you can't choose the one he does (nor would you want to... since, in the situation stipulated, it's always a goat.) The only way it can always be a goat is if the host knows which door to choose (or which doors to choose from). The host cannot always reveal a goat if he's choosing blindly.

Try the links... you'll see...