The Monty Hall Paradox--eggheads explain?

[Guy-jin]

how is this discussion going on for 7 pages? It was explained in post one.

I take it you've not taught many classes in your life.

You'll find you can explain something nine ways, and it's only on the tenth that a person finally says, "Oh, it makes perfect sense now."

 

[Calboner]

I only saw this thread after I started my own thread "Some probability puzzles" (and after Dumbcow provided a link to this one). In that thread I presented three puzzles, the third of which was the Monty Hall problem, though I presented it in a model of my own invention in order to make it less likely that people would just look up the already-published explanations of the solution. I need not have worried: even after I identified the problem and referred to the explanations of the solution on other Web sites, even after I explained the solution multiple times and in several different ways, I am still dealing with people who contend that your chance of winning is 50% whether you stay with your first guess or switch! I have felt rather lonely over there, as after a while I was not hearing from anyone but my two "doubters," Stapledshut and BoyCordoba. If anyone who has contributed elucidation in this thread, such as QuiteIrate, Guy-Jin, Manlybannisters, Rob, or Dumbcow (sorry if I've forgotten anyone), would help me out over there, I would appreciate it.

By the way, my favorite post in this thread is this one:

It is like one of those topographical maps of the moon or mars, and you can't really tell if its a crater or a mountain, it gets trippy and fucks with your mind a bit, but once you see it, you can always see it. Or like a magic eye 3D book. It doesn't make sense it doesn't make sense it doesn't make sense and then - BAM! You've unlocked the concept.

There are two ways to look at it, and I find myself jumping back and forth into each way of thinking after reading the two different thought concepts presented by each opposing party above. The paradox does make sense, you just need to unlock the concept a bit.

That, for me anyway, is the beauty — nay, the sublimity — of the problem. At first, one embraces a false solution. If one is rational and listens to good explanations of the correct solution, one begins to incline toward the correct solution. Eventually, one comes to see the cogency of the correct solution, but one is still left with the problem of why the false solution seems correct on its own terms. The final stage of comprehension comes when one understands both why the correct solution is correct and why the false solution is false — that is, one becomes aware of the false assumption that was preventing one from fully embracing the correct solution.

 

[Drifterwood]

Why are the two chances related?

By the maths, if you started with a million doors and had 999,998 opened by the host, wouldn't it make it a near certainty that you should switch on your final choice? The hare does overtake the tortoise.

 

[Phil Ayesho]

um... technically... its NOT a paradox.

A paradox is when a theoretical construct results in two mutually exclusive outcomes.

For example the nature of light- as we currently model it- is paradoxical... it is BOTH a particle AND a wave... at the same time... since one essentially excludes the other... this is evidence that our current model for light is deeply flawed.



There is nothing paradoxical about the Monte Hall problem...
Its simple and logically consistent, and true.

No paradox.

 

[Penis Aficionado]

OK, but doesn't the first pick (the one-chance-in-three pick) become irrelevant after the door is opened?

You basically throw that pick out, because you're now faced with a one-chance-in-two pick. And "not switching" is a one-chance-in-two pick just like "switching" is a one-chance-in-two pick.

In other words, the act of not switching is merely the act of picking one of two doors. And the act of switching is also the act of picking one of two doors.

It seems to me that, if you think switching increases your chances, that's because you're trying to drag the result of that first, one-in-three pick into a new situation where it doesn't belong.

 

[crescendo69]

I answered this on the later thread, but now I think I may have left out a possibility.

Here are the possible scenarios:

1) You choose the right door (with the car), then Monte shows you a goat in another door. Switching = wrong

2) You choose the right door (with the car), then Monte shows you a goat in the other door. Switching = wrong again

3) You choose a wrong door (with a goat), then Monte shows the only remaining goat in its respective door. Switching = right

4) You choose the other wrong door (with a goat), then Monte shows the only remaining goat in its respective door. Switching = right

I just knew the odds of switching and staying were even at 50%, but kept leaving out scenario #2.

 

[B_dumbcow]

It seems to me that, if you think switching increases your chances, that's because you're trying to drag the result of that first, one-in-three pick into a new situation where it doesn't belong.

That's the key to the puzzle though. The one in three pick matters completely.

You are more likely, when given the opportunity to swap, to have picked a bad choice in the first place.
So when asked whether you want to swap, you are already much more likely to be holding a goat in your door and to have the car in the remaining one.

 

[Penis Aficionado]

I see the logic of that. But the logic of the previous post is simpler: Four possible scenarios. In two of them switching wins. In two of them not switching wins. Therefore it's a 50-50 pick.

And isn't the simpler explanation usually best?

 

[B_dumbcow]

I answered this on the later thread, but now I think I may have left out a possibility.

Here are the possible scenarios:

1) You choose the right door (with the car), then Monte shows you a goat in another door. Switching = wrong

2) You choose the right door (with the car), then Monte shows you a goat in the other door. Switching = wrong again

3) You choose a wrong door (with a goat), then Monte shows the only remaining goat in its respective door. Switching = right

4) You choose the other wrong door (with a goat), then Monte shows the only remaining goat in its respective door. Switching = right

I just knew the odds of switching and staying were even at 50%, but kept leaving out scenario #2.

But...

There is a 1/6 chance of option one taking place
There is a 1/6 chance of option two taking place
There is a 1/3 chance of option three taking place
There is a 1/3 chance of option four taking place

There is a 1/3 chance of either option one or two being the scenario, because only one of three doors has a car.

And isn't the simpler explanation usually best?

They may feel like the best, but they are not always right. I've tried to explain above. The explanation is right, but the chances of each of the four scenarios happening is not equal.

 

[ManlyBanisters]

I answered this on the later thread, but now I think I may have left out a possibility.

Here are the possible scenarios:

1) You choose the right door (with the car), then Monte shows you a goat in another door. Switching = wrong

2) You choose the right door (with the car), then Monte shows you a goat in the other door. Switching = wrong again

3) You choose a wrong door (with a goat), then Monte shows the only remaining goat in its respective door. Switching = right

4) You choose the other wrong door (with a goat), then Monte shows the only remaining goat in its respective door. Switching = right

I just knew the odds of switching and staying were even at 50%, but kept leaving out scenario #2.

Scenario one and two are the same scenario - it doesn't matter which wrong door Monte shows you.

I see the logic of that. But the logic of the previous post is simpler: Four possible scenarios. In two of them switching wins. In two of them not switching wins. Therefore it's a 50-50 pick.

And isn't the simpler explanation usually best?

'Usually' doesn't enter into it with maths and probability - in this case, as in all math problems, the mathemetically correct explanation is best.

And there are not four scenarios - there are three - Please re-read dumbcow's post in the last 2 pages and my post on the top of the previous page. Honestly - this is not a trick or a question - It is a fact. Switching has a higher probability of winning you the car. It just does.

 

[Calboner]

um... technically... its NOT a paradox.

A paradox is when a theoretical construct results in two mutually exclusive outcomes.

For example the nature of light- as we currently model it- is paradoxical... it is BOTH a particle AND a wave... at the same time... since one essentially excludes the other... this is evidence that our current model for light is deeply flawed.



There is nothing paradoxical about the Monte Hall problem...
Its simple and logically consistent, and true.

No paradox.

"Paradox" has more than one meaning. The original meaning of the word, now obsolete, was "something contrary to common opinion" (as when Hamlet says, "This was sometimes a paradox, but now the time gives it scope"). Here are the glosses of now current senses in the OED:

2. a. An apparently absurd or self-contradictory statement or proposition, or a strongly counter-intuitive one, which investigation, analysis, or explanation may nevertheless prove to be well-founded or true.

b. A proposition or statement that is (taken to be) actually self-contradictory, absurd, or intrinsically unreasonable.

c. Logic. More fully logical paradox. An argument, based on (apparently) acceptable premises and using (apparently) valid reasoning, which leads to a conclusion that is against sense, logically unacceptable, or self-contradictory; the conclusion of such an argument. Freq. with a descriptive or eponymous name.
​

You are presupposing sense 2.b or perhaps 2.c, but the term "Monty Hall paradox" presupposes sense 2.a.

However, I agree with you in rejecting the term "Monty Hall paradox." The use of the term "paradox" in this context invites semantic misunderstanding and gives too much credit to so-called "intuitions" that are just prejudices and misconceptions.

I see the logic of that. But the logic of the previous post is simpler: Four possible scenarios. In two of them switching wins. In two of them not switching wins. Therefore it's a 50-50 pick.

And isn't the simpler explanation usually best?

Not when it leaves out relevant information. You assume that whenever there are two possible outcomes determined by a random process, the two must be equally probable. That is not so at all. If a process comprises two sub-processes that can lead to outcome A and only one sub-process that can lead to outcome B, and those two sub-processes are themselves equally probable, then outcome A will have probability 2/3 and outcome B probability 1/3, even though they are still randomly determined. Such is the case here. I won't repeat the explanations because you already have several lucid ones before you.

 

[crescendo69]

Scenario one and two are the same scenario - it doesn't matter which wrong door Monte shows you.



'Usually' doesn't enter into it with maths and probability - in this case, as in all math problems, the mathemetically correct explanation is best.

And there are not four scenarios - there are three - Please re-read dumbcow's post in the last 2 pages and my post on the top of the previous page. Honestly - this is not a trick or a question - It is a fact. Switching has a higher probability of winning you the car. It just does.

Because Monte always removes a wrong choice, i.e., one of the goat doors, this actually leaves only two scenarios based on your first choice.

1) If you first choose right, then switching is wrong.

2) If you first choose wrong, then switching is right.

The original odds of two out of three against you are rendered meaningless since the game is not completed yet. Only after Monte leaves you with a correct and an incorrect choice, you always choose again - right or wrong.

If Monte opened another door randomly (not based on knowledge of the correct door), then your chances of choosing right are increased to 66.6%~, since 1/3 of the time he will show you the correct door (33.3%), and the other 2/3 ot the time you must choose between the remaining doors (another 33.3%).

Can we see that Monte's knowledge and predictable behaviour actually narrows the game to two, not three scenarios? And that his random behaviour would again result in three scenarios?

I am actually enjoying this thinking process, so no hard feelings to anyone who disagrees. Love you all.:smile:

(Sorry the quotes got a little screwed up with the quoters)

 

[crescendo69]

Uh-oh, I just saw a fallacy in my "Monte picks randomly" argument.

Monte would not have a 33.3%~ chance of choosing correctly, since only two doors are left to him after you choose. Sorry for wasting your time.:redface:

 

[Calboner]

I screwed up something here (bold type added):

If a process comprises two sub-processes that can lead to outcome A and only one sub-process that can lead to outcome B, and those two sub-processes are themselves equally probable, then outcome A will have probability 2/3 and outcome B probability 1/3, even though they are still randomly determined. Such is the case here. I won't repeat the explanations because you already have several lucid ones before you.

That should have been "those three sub-processes."

 

[Phil Ayesho]

"Paradox" has more than one meaning. The original meaning of the word, now obsolete, was "something contrary to common opinion" (as when Hamlet says, "This was sometimes a paradox, but now the time gives it scope"). Here are the glosses of now current senses in the OED:

2. a. An apparently absurd or self-contradictory statement or proposition, or a strongly counter-intuitive one, which investigation, analysis, or explanation may nevertheless prove to be well-founded or true.​

This is something you often see in dictionaries... that suck....
Because its not true.
Its in dictionaries because people keep using the term to refer to something that seems contradictory, but is true.

While common usage is enough to get it in the dictionary... it doesn't make it correct usage.

For example... no matter how many times people use the term, "all told" ... they are still making a mistake.... its "all tolled".



The term paradox is technical and refers to a logical fault.
In metaphysics, you are often discussing concepts beyond physical testing... and logical analysis is the only measure of a proposition being valid, or invalid

Any intellectual concept that results in paradox is , by definition, Invalid. i.e. not possible of being true.
In philosophy... the presence of paradox proves an hypothesis false.

For example... Zeno's paradox states that motion as a continuum must be impossible... because if distance is infinitely divisible, you would have to pass thru an infinite number of points in space in a finite time... and that is impossible...
And if motion is instead a sequential series of positions, like movie frames... then in each frame nothing is actually in motion.

Now... how does this represent something that Seems inconsistent, but is nevertheless true?

Or consider the Time Traveler paradox... IF you had a time machine you could go back and kill your grandfather before you were born...

But then, you would never have been born to be able to travel back and do the murder.... so he would be alive... and you would be born...
But then you could go back and kill him... except then you would never have been born...

Again... how does this demonstrate something that is inconsistent but nevertheless true?

The whole point of paradox is that Time travel HAS to be impossible... no matter what our mathematical theories show... BECAUSE time travel results in paradoxes of causation.

The only reason the 'nevertheless true" notion creeps in is because people think that- obivously... Zeno is wrong because things clearly move...

Or that, even though light can not be a wave and a particle... it really is...

These notions simply reflect a poor understanding of science and of logic.


Sorry... light CAN NOT BE BOTH A WAVE AND A PARTICLE. we choose to look at it one way or the other... because our theories are somewhat predictive.. which means they are USEFUL.... even if they are not 'true'.

But the paradox of the nature of light PROVES unequivocally, that we simply do not have a good understanding of the nature of light.
We are making due with a useful, but knowably false model.


The same is true of Zeno... for thousands of years folks have ignored Zeno and just blithely assumed space to be a continuum... because it LOOKS like one...

However Quantum mechanics describes a granular reality, and relativity describes continua. And the primary reason we do not have a unified field theory is that ONE of these two perspectives MUST be wrong.


Zeno proved thru paradox that the one that must be wrong is the theory based upon continua. ( continua, in theory, often result in infinite expressions...and infinity can not exist.... further proof they are false )



So... while I understand that lot's of people use the term to mean something contradictory but true...

that usage is actually the precise Opposite of what the term means... it means something that CAN'T be true BEcause of contradictions.

A dictionary entry that states that a word can mean it exact opposite is a flawed entry.











Thus Zeno proves that space MUST be granular and can not be a continuum...​

 

[Calboner]

This is something you often see in dictionaries... that suck....
Because its not true.
Its in dictionaries because people keep using the term to refer to something that seems contradictory, but is true.

While common usage is enough to get it in the dictionary... it doesn't make it correct usage.

For example... no matter how many times people use the term, "all told" ... they are still making a mistake.... its "all tolled".

The term paradox is technical and refers to a logical fault.

While I agree with you that dictionaries can make mistakes, and that usages that are common can nevertheless be, relative to certain considerations, incorrect, you are wrong several times over in this particular case.

First, the dictionary that I cited was the OED — the Oxford English Dictionary, the greatest historical dictionary ever produced. That does not mean that it is infallible, but it does mean that anyone who describes it as a "dictionary that sucks" either is not speaking seriously or doesn't know what he is talking about.

Second, your particular illustration of commonly accepted error is itself an error. "All told" comes from the use of "tell" to mean "count." You can verify this historical fact here (The Word Doctor), or here (The American Heritage Guide to Contemporary Usage and Style), or in the OED itself (entry for "tell").

Third, there is just as much historical ground for regarding the meaning that you attribute to the word "paradox" as incorrect as there is for so regarding the meaning or meanings that you object to, as all those meanings are deviations from the original meaning of the word. That meaning, in Greek (from which the word derives), from Latin, from French (from which it came into English), and originally in English, is the first one that I cited — "contrary to received opinion."

Fourth, even if the word has been given a technical meaning in certain fields, that technical meaning has no authority over common usage because the word is not of technical origin. When a word is of a technical origin, like "deconstruct" (originated by the pseudo-philosopher Jacques Derrida), there is reason to object to the distortion of its meaning when it is taken up in popular language (and especially when the term was pretentious and obscure to begin with, like that one). But this is not such a case. "Paradox" was a term in common usage before it was a technical term.

As I said earlier, I share your aversion to the phrase "Monty Hall paradox." And I can understand why you would consider it desirable to confine the use of the word "paradox" in the way that you suggest, as it is better to have a word with one clear meaning rather than a loose bundle of different meanings. But the fact is that those meanings are at least as well founded in current and historical usage as the one that you favor.