Monty Hall Problem

[articulett]

Claus, it says what I said. Your odds of the car being behind the first door you chose remain at 1/3. Always.

Herzblut-- I stopped reading you long ago... I cannot make sense of your tangents.

Claus you are wrong... and you are too stubborn and daft to admit it. Are you actually claiming that at some point the odds of you having the car behind the door you chose is more than 1/3? If you are saying something other than this, you've communicated it poorly. If this is what you really think... go check with someone you find intelligent... find one intelligent source that will agree that your odds of your original door being the correct door is ever more than 1/3. (You are just so embarrassingly wrong so often that it amazes me that you imagine yourself a skeptical expert. You aren't making sense to anyone but yourself, I suspect.) No matter what you believe or how much you believe it-- in the Monty Hall scenario the odds of the car being behind the first door you chose is NEVER more than 1/3. Never.

Imagine 10 cars. Your odds of having chosen the correct car is 1 in 10-- those odds never get better. If they reveal a goat, your odds go up by switching to 1 in 9-- if you keep your same door-- you still have 1 in 10 chance of being right.

Really. Ask someone whom you know to be smart.

And I accept your apology. Your bizarre attempts to derail threads with trying to prove me wrong at every turn only makes you look increasingly foolish.

 

[ddt]

Here we disagree. It may be because of a subtlety of the language, or maybe just an ambiguity. My interpretation of the "he reveals behind another door that there is a goat" clause is that it presents two facts: Monty opened another door and there was a goat behind the other door.

As I interpret it, the "he reveals ... that" contains the intent to reveal a door-with-goat.

Are two assumptions (Monty always opens a door and it's always a goat door) more minimal and reasonable than just one assumption (Monty always opens a door)?

Is that a rhetorical question?

I am a native English speaker, so you have me at a disadvantage.

As Herzblut, I'm not a native speaker, so I have the same advantage of being able to claim things about English and afterwards plead ignorance

As written, it makes uses the present tense in a clumsy way. Is it present tense with future meaning (these are things that must be) or present tense with past meaning (these are things that happen to be)?

I favor the latter interpretation, but I can understand why others might prefer the former.

That's a nice categorization. Note that, for instance, kitchen recipes aren't put in future tense either. And the actions in a recipe aren't chance occurrences either - they have to be done, if you want to arrive at the goal. In the same vein, I've never seen puzzles stated in the future tense either: if you want to arrive at the solution, you have to take the actions of (possibly) deterministic agents as rules.

The player will pick a door, then Monty will open one of the two other doors that will reveal a goat, and then Monty will offer the player the option to switch to another door of his choice.​

Here, I think with everything in future tense, the ambiguity is gone.

I actually had begun to write a post with all kinds of possible formulations of the problem, without changing tenses. Let's strip this one of the future tense:

The player picks a door. Then Monty opens another door that has a goat behind it. Then Monty offers the player the option to switch to another door.​

To me, the intent of revealing a door with a goat - opposed to the door with the car - is signified by the use of a restrictive relative clause. It may not be 100% clear that he will in all cases open a door, but if he opens one, it's one with a goat. Compare with:

The player picks a door. Then Monty opens another door, which has a goat behind it. Then Monty offers the player the option to switch to another door.​

which has a non-restrictive relative clause. Now it's a chance occurrence that the door he opens has a goat behind it. A Brit might use "which" in the first case too, so then it boils down to a comma what's meant.

Now, let's try it all in past tense:

The player picked a door, then Monty opened one of the two other doors that revealed a goat, and then Monty offered the player the option to switch to another door of his choice.​

With everything in the past tense, is it clearer that Monty's motives may have a role in the analysis? I am hoping so.

I absolutely concur with that.

Unfortunately, the original problem statement wasn't at all clear. The use of the present tense gives in characteristics of both the future and past tense versions. And, as I said, I favor the past tense interpretation, but your actual mileage may vary.

That begs another question. If you were to formulate the puzzle, would you favour using the future tense for everything, or would you favour keeping it in the present tense and add some words to signify that these are really rules that have to be followed by the game host?

Just focusing on the part which door has to be opened by the host (and thus not the part whether he opens a door at all), I think the original formulation by vos Savant:

and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat.
​

by adding the "knows what's behind the doors" signifies the intent of Monty to open a door with a goat behind it.

 

[Herzblut]

Claus you are wrong... and you are too stubborn and daft to admit it. Are you actually claiming that at some point the odds of you having the car behind the door you chose is more than 1/3?

Well. What about the odds of you having the car behind the door at the point when you've opened the door to find that you are having the car behind the door?

Gosh! This post shows the most stupid ignorance I've ever seen.

 

[sol invictus]

Claus, it says what I said. Your odds of the car being behind the first door you chose remain at 1/3. Always.

Maybe you're thinking of the odds before Monty opens a door? Everyone agrees that's 1/3, but once a door has been opened the odds (that the car is behind your door) can certainly change.

Imagine Monty opens both of the other doors - clearly then the odds have changed, right?

 

[ddt]

A funny thing about humans is that they tend to go with their first choice--

On this and other psychological aspects of the problem, see this paper that is referenced from the wiki page.

 

[articulett]

Herzblut... your posts show the most pompous ignorance I've ever seen. You keep changing the original scenario to make others wrong.

I think all the smart people understand it is a question about odds. After you win the lottery you have a 100% chance of your numbers being drawn too. That doesn't change your odds of you having picked the right numbers in the first place. The model is not about post hoc reasoning... it's about odds. Yes, if I flip a coin and it ends up heads... then the 50/50 probability that it will end up heads becomes 100%. But that's a silly game for the silly to play... as are all your rewritten scenarios where Monty reveals the car and so forth. They are tangential and off topic.

The problem is often used to show how people think irrationally. Most people think like Claus... that your odds suddenly change when there's 2 doors left. However, that is never the case. The odds of your having chosen the correct door is 1 in 3 in the scenario as described. The question is should you switch? Your bizarre mental games are probably why few people engage you in dialogue.

 

[articulett]

Maybe you're thinking of the odds before Monty opens a door? Everyone agrees that's 1/3, but once a door has been opened the odds (that the car is behind your door) can certainly change.

Imagine Monty opens both of the other doors - clearly then the odds have changed, right?

No. Your odds don't change. If he opens both doors you suddenly know with 100% certainty where the car is. But the game as described do not up your odds of having chosen the correct door in any way. It's counter intuitive, but worth understanding.

Think of a situation with 9 goats and one car. Certainly you odds of choosing the right door the first time are 1 in 10... this doesn't get any better when he shows you a goat... He just eliminates a losing entry... that doesn't make you more likely to be a winner. However switching ups your odds because now there only 8 goats and one car... it's 90% likely that the car is amongst those 8 goats. So instead of the 10 percent chance you had, you now have slightly more than an 11% chance by switching... not the best odds... but better than 10%.... If he eliminates another goat your odds go up to almost 13% by switching.

There's all sorts of programs on line-- applets etc. where you can prove this to yourself...

But your odds of having picked the right door the first time, do not change-- the elimination of goats just increases the chance that it's in your favor to switch. But statistically it's always in your favor to switch.

I'm glad to explain this to people who want to understand. It is true whether you believe it or not. But if no amount of evidence will change your mind, then the loss is yours. Like an optical illusion, it's important to understand how your perceptions can be wrong. It's an important part of critical thinking, in fact.

 

[sol invictus]

No. Your odds don't change. If he opens both doors you suddenly know with 100% certainty where the car is. But the game as described do not up your odds of having chosen the correct door in any way. It's counter intuitive, but worth understanding.

articulett, I've never been even slightly confused by this problem. I understood it immediately the first time I read about in the paper years ago (and was shocked by the resulting public debate, part of which I attributed to sexism - but that's another story). Probability and statistics, specifically conditional probabilities, are a big part of the work I do every day.

Think of a situation with 9 goats and one car. Certainly you odds of choosing the right door the first time are 1 in 10... this doesn't get any better when he shows you a goat... He just eliminates a losing entry... that doesn't make you more likely to be a winner. However switching ups your odds because now there only 8 goats and one car... it's 90% likely that the car is amongst those 8 goats. So instead of the 10 percent chance you had, you now have slightly more than an 11% chance by switching... not the best odds... but better than 10%.... If he eliminates another goat your odds go up to almost 13% by switching.

That depends on the rules governing Monty's behavior.

Please take a moment and consider a simpler situation. You pick a door (out of 3) at random. At that point, we all agree the odds that the car is behind it are 1/3. Now suppose Monty opens both of the other doors, revealing goats. What are the odds now?

The odds the car is behind the door you picked can certainly change when Monty opens more doors. In the classic Monty hall problem they do not - because of the particular rules Monty follows (he knows where the car is and he will never open your door or the door to the car). But this is by no means guaranteed, as the last several pages of this thread - or my example above - show.

 

[articulett]

There are all sorts of variations on this...
Here's one-- you give two people two coins and ask them to flip them...
You then ask person A whether he has any heads, "she says, yes--in fact the first coin I flipped was a head". The second person says. "Yes". What are the odds that the first person has 2 heads? What are the odds that the second person does? The odds are not equal, I'll tell you that. Which person do you think is more likely to have 2 heads?

 

[CFLarsen]

The problem is often used to show how people think irrationally. Most people think like Claus... that your odds suddenly change when there's 2 doors left. However, that is never the case. The odds of your having chosen the correct door is 1 in 3 in the scenario as described. The question is should you switch? Your bizarre mental games are probably why few people engage you in dialogue.

That's precisely what happens: The odds do change, because the situation has changed: You suddenly know something you didn't know before.

Did you read the article I linked to?

 

[articulett]

No sol-- what's changing is whether you are going to switch doors or not. If he opened both doors, then the riddle is over-- everyone knows exactly where the car is and whether the person should switch. But that never made the original choice better than 1 in 3-- if he shows 2 goats-- it just so happens that the 1 in 3 chance turned out to be the winner. If he shows the car-- that shows the 2/3 chance that switching was right. 1/3 of the time in this new scenario, he'll end up showing you 2 goats. 2/3 of the time, he'll be showing you a goat and a car.

It's not a sexist thing. It's basic math that anyone could do. It's just that some men feel so certain that they are right about this that they cannot imagine that they've fooled themselves. When you change the scenario, you aren't changing the odds of your first door being right!

This has been demonstrated countless times but there are still some people -- always men-- who insist it cannot be so... but it is so. Your odds of having chosen the correct door the first time do not change-- just the information...

I won't argue this. Feel free to ask any expert... read the wikipedia entry... design your own damn program... Facts are facts

And in the coin scenario... the first person is more likely to have 2 heads (50/50)
The second person has a 1 in 3 chance of having 2 heads. Their tosses were either: HT, TH, or HH. See? It's very similar with the Monty hall problem. You are just thinking about it incorrectly-- but it's a very common mental mistake. There are tons of books on this. Predictably Irrational by Dan Ariely is one I'm reading currently. Don't let your ego get in the way of understanding this. It will be your loss.

Thomas Kida's book, Don't Believe Everything You Think is also good. Skepdic's dictionary has little lessons on these things. The truth doesn't change just because you don't understand it.

 

[articulett]

That's precisely what happens: The odds do change, because the situation has changed: You suddenly know something you didn't know before.

Did you read the article I linked to?

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. Your information changes... the odds of your original choice being the correct one-- do not. Your odds of having picked the correct door the first time remain 1 in 3-- even after he shows you a goat.

There was always a 2/3 chance that you did not have the car-- when he shows you the goat-- as he always can-- that means the 2/3 chance is on the remaining door.

Did YOU read the article??

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.

 

[Herzblut]

you give two people two coins and ask them to flip them...
You then ask person A whether he has any heads, "she says, yes--in fact the first coin I flipped was a head". The second person says. "Yes". What are the odds that the first person has 2 heads?

50%?

What are the odds that the second person does?

1/3?

This is well known and has also been discussed in this thread.

 

[sol invictus]

No sol-- what's changing is whether you are going to switch doors or not. If he opened both doors, then the riddle is over-- everyone knows exactly where the car is and whether the person should switch.

In other words, the odds have changed.

But that never made the original choice better than 1 in 3-- if he shows 2 goats-- it just so happens that the 1 in 3 chance turned out to be the winner.

No one is disputing that the odds are 1/3 before Monty opens any doors! What is at issue here is your claim that the odds do not change after he opens the door or doors. That's just not true (in general).

It's not a sexist thing. It's basic math that anyone could do. It's just that some men feel so certain that they are right about this that they cannot imagine that they've fooled themselves.

There certainly was some degree of sexism involved in the response to vos Savant's column - see here, especially the last comment. That's what I was taking about. However you are wrong that only men claim the odds can change. In fact, vos Savant herself says the odds change when Monty (the gameshow host) doesn't know which door the car is behind.

When you change the scenario, you aren't changing the odds of your first door being right!

For the third time, no one is arguing about what the odds were originally, before Monty did anything.

This has been demonstrated countless times but there are still some people -- always men-- who insist it cannot be so... but it is so. Your odds of having chosen the correct the first time do not change-- just the information...

Always men? Really?

If the host is clueless, it makes no difference whether you stay or switch. If he knows, switch

If the host is clueless, the odds the car is behind the original door have changed from 1/3 to 1/2.

I won't argue this. Feel free to ask any expert... read the wikipedia entry... design your own damn program... Facts are facts

I am an expert, articulett.

And in the coin scenario... the first person is more likely to have 2 heads (50/50)
The second person has a 1 in 3 chance of having 2 heads. Their tosses were either: HT, TH, or HH.

Correct.

It's very similar with the Monty hall problem. You are just thinking about it incorrectly-- but it's a very common mental mistake.

Please take a step back and think again, slowly. No one is disputing that the odds are initially 1/3. No one is disputing that they stay 1/3 in the standard problem. No one is disputing that switching gives you a 2/3 chance of winning, again in the standard setup. However, your claim that the odds never change - no matter what Monty's rules are - is simply false.

 

[Michael C]

Articulett: the odds can change when Monty opens a door, depending on what new information can be inferred from what is revealed behind the door. CurtC summed up the different possibilities here: http://www.internationalskeptics.com/forums/showpost.php?p=3906664&postcount=418

Marilyn vos Savant (who is known to be very smart indeed and an expert on this particular problem) also explained the difference between the version where Monty chooses a door at random and the original problem, where he knows where the car is. The odds really do change to 50/50 when Monty opens a door at random, happening to reveal a goat. In the original version, where Monty always reveals a goat because he knows where the car is, the chance of the prize being behind your chosen door stays at 1/3, while that of it being behind the remaining door is 2/3.

You can read this in the Wikipedia article, but for convenience I'll quote vos Savant again here. The question is: is it in the contestant's interest to switch doors after the host has revealed the goat? The answer is: it depends on what information the host had when choosing the door. As Marilyn vos Savant puts it:

If the host is clueless, it makes no difference whether you stay or switch. If he knows, switch

 

[articulett]

Wrong, Sol. Ask any expert you know... run a program. Your odds of your first choice being the winner are always 1 in 3 in the Monty Hall problem.

Even if he reveals both doors every time-- You will still only have chosen the car on your first choice 1/3 of the time. You still will switch to get the car 2/3 of the time. Of course, the more he does this the more greater you see these probabilities.

The odds that you chose the right door on your first pick are always 1 in 3-- through the whole game. Sometimes that 1 in 3 will pay off-- in fact 1/3 of the time it will.

In your new scenario the odds never change. 1/3 of the time, he'll be revealing 2 goats and you'll have chosen the correct door. 2/3 of the time he'll show you a goat and a car and you will change.

The greater the numbers of tests in this scenario the easier you'll see this. In the original scenario you have a 2/3 chance of winning by switching. You have a 1/3 chance of winning by staying with your first choice.

You can try to convince yourself otherwise, but these are well understood odds.

YOUR ODDS OF HAVING CHOSEN THE RIGHT DOOR THE FIRST TIME DO NOT CHANGE THOUGHOUT THE GAME. It does not suddenly go up to 50% because he's eliminated one of the doors.

 

[Skeptical Greg]

Knowing there are 4 aces in a deck of cards, does not change the odds of one of them being dealt. Stacking the deck does..


The host ' knowing ' where the car is does not change the odds.. The steps taken because of his knowledge does..

 

[hgc]

Can't believe this thread lives on after all these years. I thought the question was settled on about page 3.

Go ahead and keep flailing, articulett. You'll come to understand eventually. It's inevitable.

eta: Huh? I don't even understand what anyone's aguing about anymore. What are you claiming, articulett?

 

[sol invictus]

Wrong, Sol. Ask any expert you know... run a program. Your odds of your first choice being the winner are always 1 in 3 in the Monty Hall problem.

Please be aware you're not only disagreeing with every poster in this thread, but with Marilyn vos Savant herself.

Even if he reveals both doors every time-- You will still only have chosen the car on your first choice 1/3 of the time. You still will switch to get the car 2/3 of the time.

You're simply not understanding the situation. Again, suppose Monty opens both other doors, and both have goats behind them. What are the odds that the car is behind your door?

Note that I am not asking you how often this situation will occur if the game is played many times - I'm asking about a particular class of instances, where both other doors are revealed to have goats behind them.

In your new scenario the odds never change. 1/3 of the time, he'll be revealing 2 goats and you'll have chosen the correct door. 2/3 of the time he'll show you a goat and a car and you will change.

That was not my scenario. You're arguing with yourself here.

YOUR ODDS OF HAVING CHOSEN THE RIGHT DOOR THE FIRST TIME DO NOT CHANGE THOUGHOUT THE GAME. It does not suddenly go up to 50% because he's eliminated one of the doors.

Then you should write to vos Savant and tell her she is wrong about the "clueless" case.

 

[CFLarsen]

Yes Claus, I read it. I do this demonstration in my class every year

Seriously?

If you tell them what you have argued here, then you are demonstrably miseducating your students.

Did YOU read the article??

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.

Sorry, but you are the one who is wrong here. Not because you are of a certain gender, but simply because you are wrong - period.

I sincerely hope that you don't in reality single out a gender in your classroom, deriding them not just for being wrong (although you are in fact the one being wrong), but being wrong because they are of the opposite sex than you.

What you are describing here is not in any way an acceptable educational method. You are not just teaching them false logic, you are teaching them that they are wrong because they are not women.