The monty fucking hall problem.

[Cyrene]

Heuristics and biases teach us that humans systematically make errors in reasoning, these errors are common in lay people and scientists alike.

The monty hall problem:

Theres a gameshow you're on, theres three doors (one with a car, and two with goats). you choose door1. Monty opens door 3 to show you a goat.

What are the chances you win the car if you choose door1 (your door) or door2 (the unopened) door. Most people think its either 1/3 (with either door) or (1/2 chance) with either door. But its not. Door1 is 1/3 chance to win. door2 is a 2/3 chance to win.

I came across this problem and I actually solved it. Which isn't surprising because its a brutally simple reason why door2 is a 2/3 chance to win, but because every single person i've talked to since, is convinced* that its the same probability with either door.

[Cyrene]

The reason is because you're dealing with different subsets and the goat reveals information which was previously unknown. (this is all only true if the gameshow host knows the location of the car, btw.)

2/3 times the car is going to be in the two doors that you didn't pick (by chance) so that when the goat is revealed, picking the *other door, has a higher probability to win.

If anyone is doubtful of how in turns out in practice, grab a deck of cards, (use an ace for the car and 5's for the goats)

I pass you one card face down, and I have two cards, (2/3 times i'll have the ace) I show you a 5 and all thats left is two face down cards. 2/3 times, i'll have the acec.

[TheLizardKing]

to put it simply the reason the second door is 2/3 is because monty knows where the car is and the fact that he decided to show you the goat in the 3rd doors means it is more likely there is not a goat in the 2nd door. the way you worded the problem in the first post fails to mention that monty knows where the car is and deliberately showed a goat, without this information I assumed that monty chose to open the 3rd door without knowing where the car was, in which case the two remaining doors would have an equal chance 1/2. I don't think this is a problem in human reasoning but more like a lack of information and failure to properly state the problem or perhaps my failure to assume the actions of monty were deliberate, either way it's kind of a stupid problem, at least in my opinion

[shorelinetrance]

edit//nevermind i'm dumb.

[Wonderer]

is this the problem from the movie 21?

here's how that one goes down

there are 3 doors and one winner

your only choice is to pick a random door.

you pick door 1 lets say.

the host says, but wait a second, and reveals door 3 to be empty.

next he offers you a chance to switch your answer from door 1 to door 2.

the question is why is that a good idea.

the answer put simply is this.

when the game show started there was a 1/3'd chance of the winning door being door number 1, and a 2 third chance of it being door number 2 or 3.

with the disposal of door 3, door 2 now represents the probability of it being in door 2 or 3 which is higher than it being door one.

variable of change or something it is called.

[shorelinetrance]

if the gameshow host knows whats behind the doors, switch.

if he doesn't know, it's the same odds.

[Uccisore]

The part that I have a hard time with is, what if we add another contestant or viewer partway through?

Suppose the scenario all plays out just as described in the example, and after that I turn on the TV and start watching. For me, I see 2 doors and a goat. From the two of them talking, I get the general idea that there's a car behind one of the two doors, and a goat behind the other. It seems like in that situation, I'd have a 50/50 chance of picking the right door, if I didn't know which one the contestant picked originally. Right?

Can someone explain to me how discovering which door the contestant picked would provide information that makes one door more likely than the other? What is it about the contestant picking door 1 that makes me go, "Ah, it's probably door 2!"

[IrvingWashington]

Has anyone read 'The Man Who Only Loved Numbers'? Some good stories in there about how some of the finest mathematical minds of our century couldn't grasp this. For my part, I understand the proof I saw, but its still very counter-intuitive. If Paul Erdos didn't understand it, I don't feel so bad.

[Tab]

How about looking at it this way..?

Three doors: 2 goats, one car - you choose one. 66% you chose a goat.

ie: You prolly did.

So, gameshow guy is left with two doors. According to the above there's a 66% chance you've left him with a door with a car, and a door with a goat.

Obviously, he can't open the door with the car. Cos he'd get fired. So, 66% of the time, he has no choice about which door to open.

Ergo, if you choose the door the gameshow host didn't open - you'll get the car, 66% of the time.

Not bad odds really - 2 in 3, certainly better than the 1 in 3 you started with.

Not so unintuitive if you look at it backwards.

[someoneisatthedoor]

I understand the mathematical proof, I just place no stock in mathematical proofs concerning probability.

Likewise, a lot of people do this intuitively. As such, it isn't that they 'don't get' the answer, it's that they don't believe in it.

[Tab]

I understand the mathematical proof, I just place no stock in mathematical proofs concerning probability.

??? You'll hafta 'splain the last bit.

Likewise, a lot of people do this intuitively. As such, it isn't that they 'don't get' the answer, it's that they don't believe in it.

Fair enough, sometimes common sense ain't enough, but then that's what makes aquiring uncommon sense all the more worthwhile, no..?

[Uccisore]

I understand the mathematical proof, I just place no stock in mathematical proofs concerning probability.

Likewise, a lot of people do this intuitively. As such, it isn't that they 'don't get' the answer, it's that they don't believe in it.

Be simple enough to test with 2 people and a deck of cards, wouldn't it? The stats aren't just numbers in space, if the example is right, switchng really should get you the car (or ace) 66% of the time.

[Cyrene]

someoneatwhatever is always making absurd and nonsensical claims that he doesn't even try to justify. Of course his beliefs are irrelevant, its not just numbers in space, which anyone intelligent enough to operate a deck of cards should be able to find out for themselves. His beliefs won't alter any probability when making the choices or the statistical outcome. Don't expect any coherency besides complaints about my repeated use of the word 'absurd' sad and pathetic.

people don't believe it only because they don't 'get' it. It works out that way in the real world, regardless of personal opinion. thier lack of belief doesn't change the probability difference between picking thier card or the non revealed. if they don't believe, they don't get it, because thats just how it works out. again master the playing card and get back to us.

[Tab]

P'raps we should give SIATD the benefit of the doubt... Maybe he's operating in an alternate reality, philosophically at least.

[Impenitent]

siatd makes perfect sense to me

an unjustified prediction is an unjustified prediction

-Imp

[realunoriginal]

How can a prediction be "unjustified"? It's either going to turn out accurate or inaccurate between subjective opinions based upon the criteria it self-describes.

Men only gather around the predictions that keep proving themselves true, which is the error of taking science as a faith...

[Impenitent]

it begs the question or it does not.

invalid assumptions are not justified.

-Imp

[realunoriginal]

it begs the question or it does not.

invalid assumptions are not justified.

-Imp

Justice is ambiguous then ... it always gets stuck in hypothetical contexts.

But isn't that what all predictions are, hypothetical? (until they manifest) So then -- what is an "invalid assumption" in context to the hypothetical?

This is interesting, because this logic seems flawed to me when it can't be practical in any sense whatsoever.

[Impenitent]

it begs the question or it does not.

invalid assumptions are not justified.

-Imp

Justice is ambiguous then ...

no, justice is whatever the guy with the gun says it is.

it always gets stuck in hypothetical contexts.

But isn't that what all predictions are, hypothetical?

the hypothetical is never true

(until they manifest)

So then -- what is an "invalid assumption" in context to the hypothetical?

if it begs the question it is invalid

This is interesting, because this logic seems flawed to me when it can't be practical in any sense whatsoever.

no, logic itself is flawed.

-Imp

[realunoriginal]

no, justice is whatever the guy with the gun says it is.

That's exactly the definition of ambiguity -- whatever a person is compelled to do is whatever a person is compelled to do.

The situation says nothing until beforehand or afterward in context, until it is discussed and made sense of, otherwise we're just being. Actions speak louder than words, yes, but what are we doing here on this forum -- acting or writing? Where do word acts begin and end? Is this even the problem at hand?

the hypothetical is never true

The hypothetical is a memory of something that may or may not be real, according to fiction or nonfiction. Predicating events based on hypothetical contexts are what men instinctively do in order to anticipate the exchange of gunfire. The hypothetical can be true when what is "true" is definitely going to happen (like that I'm going to drive my car within the next two hours).

I want some clarification...

if it begs the question it is invalid

Skepticism & Nihilism continually beg the question. Pragmatism ends it, so it also forces men to decide what is "valid" and what is "invalid".

This is interesting, because this logic seems flawed to me when it can't be practical in any sense whatsoever.

no, logic itself is flawed.

-Imp

It's only as flawed as flawed men allow it to be. The fallacy of logicians (and most other logically predicated sciences) is that they don't know how, where, or when to update their language after mistaking it as an absolute authority. I won't speak for others, but my authority rests in the same place I put my faith...

[Impenitent]

no, justice is whatever the guy with the gun says it is.

That's exactly the definition of ambiguity -- whatever a person is compelled to do is whatever a person is compelled to do.

no, that's not ambiguous, that's power

The situation says nothing until beforehand or afterward in context, until it is discussed and made sense of, otherwise we're just being. Actions speak louder than words, yes, but what are we doing here on this forum -- acting or writing? Where do word acts begin and end? Is this even the problem at hand?

not at all, but you brought up justice

the hypothetical is never true

The hypothetical is a memory of something that may or may not be real, according to fiction or nonfiction. Predicating events based on hypothetical contexts are what men instinctively do in order to anticipate the exchange of gunfire. The hypothetical can be true when what is "true" is definitely going to happen (like that I'm going to drive my car within the next two hours).

you could have died in the interim. the future is never definite. that is the error.

I want some clarification...

the future is always unknown and "probability" is based on begging the question. the fact that it has always happened that way in the past does not insure/ensure the future event will occur... to claim it must is not valid logically as it begs the question and is an error...

if it begs the question it is invalid

Skepticism & Nihilism continually beg the question. Pragmatism ends it, so it also forces men to decide what is "valid" and what is "invalid".

pragmatism does nothing of the sort. simply proclaiming that the error will be ignored does not make the error go away...

This is interesting, because this logic seems flawed to me when it can't be practical in any sense whatsoever.

no, logic itself is flawed.

-Imp

It's only as flawed as flawed men allow it to be.

as an invention of men

The fallacy of logicians (and most other logically predicated sciences) is that they don't know how, where, or when to update their language after mistaking it as an absolute authority.

no, they continue the lie of authority when they haven't got a logical leg on which to stand.

I won't speak for others, but my authority rests in the same place I put my faith...

as a question of faith, it is no more valid as a belief in any god...

-Imp

[Cyrene]

Ihate my threads degrading to this drivel. This is a testable prediction, people who don't believe simply can't wrap thier heads around it, as the same statistical results or close enough happens consistently in the real world WHEN TESTED. leave the BULLSHIT run of the mill arguments against science/statistical predictions until you can show that in the real* world they don't routinely come true.

again i suggest a deck of cards for those people too incompetent to do the mental math if you don't 'get' it grab some cards if the people in question thinks its faith 200 rounds with a deck will prove them wrong, the correct card *will be in the dealer's hand consistantly enough to say the prediction is correct.

seriously not all my threads need to be dragged down by philosophy based on the standards of a seven year old. real world ideas have testable predictions, the worth of claims depends on the accuracy of those predictions, not any mental midget's ideas about thier worth...

[Mr Reasonable]

What about this Cyrene?

http://en.wikipedia.org/wiki/Newcomb's_paradox

Is this philosophically related to your point?

I see it as meaning that there is alot more difficulty to determining probability than it might seem when we just think of dividing fractions.

Another guy I've been reading thinks that it's just a prisoner's dilemma cut in half.

Game theory rules.

[Reality Check]

Here's what I consider to be the most easily grasped explanation for this problem:

Assume that there are ten doors to choose from to begin with, with goats behind nine of them and a car behind only one.

You pick one of the ten doors. Your chance of selecting the door with the car behind it is 10% and the chance that the car is behind one of the other nine doors is 90%.

Then the game show host, who knows which door the car is behind and will not open that door, opens eight of the remaining nine doors to reveal goats behind them. In no way does the fact that eight of the other nine doors have now been opened to reveal goats behind them affect the 10% chance that you selected the correct door to being with. It doesn't affect it, because YOU ALREADY KNEW THAT AT LEAST EIGHT OF THOSE NINE DOORS HAD GOATS BEHIND THEM!

You are then given the option of keeping the door you picked or of switching to the only other unopened door.

Since it is obviously 90% likely that the car is behind the sole remaining unopened door that you didn't pick and only 10% likely that it is behind the door that you did pick, only an idiot would keep the door that he originally picked, right?

Well, the same principle applies whether there are three doors to choose from to begin with or ten or a million. The only difference is, is that the odds that you selected the door with the car behind it to begin with obviously get much worse the more doors that there are to choose from. This also means of course that the odds that the car is behind one of the doors that you didn't select get much better.

[realunoriginal]

I believe the point that Cyrene is making is how cognitive probabilities are processed in more basic and "stupid" people. On a game show, for the host to pick all the incorrect doors and leave the final "chance" at 50-50, then people misinterpret that the odds of picking the correct door ends up as a 50% chance. In reality, those who are more intelligent see through the "game" of the "show" ... that they detect that the host is misleading the contestant and the viewers (conceptually) by a "sleight of mind". In the end, it appears as the pick comes down to a 50% chance, when in reality, the chance was very low to begin with (depending on the amount of doors w/goats behind them).

The "problem" here is how the probability changes in a real-time context. The human mind works cognitively in this manner (compared against an intelligent or ignorant person) by anticipating possible changes and adhering expectations toward them. Most people cannot account for this, because they are literally stupid. However, a more intelligent person will pick up on the clues and eventually understand that the door situation stayed consistent in context from the beginning to the end and the "game show" is a mere illusion to those who are fooled by it.

What would really throw such an "intellect" off is if the game show host randomly picks the correct door from time-to-time rather than leaving it to the last. Then, the odds become randomized again in expectation only ... and the "game" loses its appeal (of anticipation of coming down to the final two doors rather than seeing the result straight-out).