Probability: Monty Hall Problem in the movie '21'

OutlawPete

(Don't watch video or read any other posts until AFTER you have voted)

You're on a Game Show:

You're given the choice of one of three Doors.

One door has a new Car behind it and the other two: Goats.

You pick Door No.1.

The host (who knows what’s behind all the doors) opens Door No. 2 and says:

"Do you want to stick with your decision of Door No. 1 or do you want to change your choice to Door No. 3?".

What would you do, stick with your choice (Door No. 1) or would you switch (Door No. 3)?

https://www.youtube.com/watch?v=Zr_xWfThjJ0

I have always considered the 'Monty Hall' problem to not be a "probability" problem at all (in the way it is sometimes presented at least) but more that it is in fact a 'trick question' - which is why I feel was the reason that even some mathematicians have historically got the promlem (riddle?) wrong.

I have changed my mind back and forth on the problem (I most likely will again ) but what says you.

sponsoredwalk

I thought it was a trick question as well, or dependent on something outside
of mathematics, until I read the following explanation.

Cars and goats: the Monty Hall dilemma
On Sunday September 9, 1990, the following question appeared in the
“Ask Marilyn” column in Parade, a Sunday supplement to many newspapers
across the United States:

Suppose you’re on a game show, and you’re given the choice of three
doors; behind one door is a car; behind the others, goats. You pick a
door, say No. 1, and the host, who knows what’s behind the doors,
opens another door, say No. 3, which has a goat. He then says to you,
“Do you want to pick door No. 2?” Is it to your advantage to switch
your choice?—Craig F. Whitaker, Columbia, Md.

Marilyn’s answer—one should switch—caused an avalanche of reactions,
in total an estimated 10 000. Some of these reactions were not so
flattering (“You are the goat”), quite a lot were by professional
mathematicians (“You blew it, and blew it big,” “You are utterly incorrect .
. . . How many irate mathematicians are needed to change your mind?”).
Perhaps some of the reactions were so strong, because Marilyn vos
Savant, the author of the column, is in the Guinness Book of Records for
having one of the highest IQs in the world.

The switching question was inspired by Monty Hall’s “Let’s Make a Deal”
game show, which ran with small interruptions for 23 years on various U.S.
television networks.

Although it is not explicitly stated in the question, the game show host will
always open a door with a goat after you make your initial choice. Many
people would argue that in this situation it does not matter whether one
would change or not: one door has a car behind it, the other a goat, so
the odds to get the car are fifty-fifty. To see why they are wrong,
consider the following argument. In the original situation two of the three
doors have a goat behind them, so with probability 2/3 your initial choice
was wrong, and with probability 1/3 it was right. Now the host opens a
door with a goat (note that he can always do this). In case your initial
choice was wrong the host has only one option to show a door with a
goat, and switching leads you to the door with the car. In case your initial
choice was right the host has two goats to choose from, so switching will
lead you to a goat. We see that switching is the best strategy, doubling
our chances to win. To stress this argument, consider the following
generalization of the problem: suppose there are 10 000 doors, behind one
is a car and behind the rest, goats. After you make your choice, the host
will open 9998 doors with goats, and offers you the option to switch.
To change or not to change, that’s the question! Still not convinced?

link, page 4

I doubt you could show how that this is not a question of probability,
makes perfect sense to me as a question of probability & the fact there
are so many variants on this problem illustrates the importance of
probability.

Of course it's a trick question :cool: The movie got it right, I don't see how
there is NO advantage, you haven't shown us why that is.

OutlawPete

No

sponsoredwalk wrote: »

Of course it's a trick question :cool: The movie got it right, I don't see how there is NO advantage, you haven't shown us why that is.

Hhhm, I guess I am presuming that if his initial guess is correct and he switches, he loses - but of course, that doesn't make it (mathematically) wrong to switch.

Real mindfcuk, but yeah - you're right

sponsoredwalk

Yeah I mean there could be a sheep, or a lion or whatever behind the other
door, before I read this I really didn't think it was smart to switch
It is a mindfcuk alright but certainly makes probability look more interesting
as a subject.

OutlawPete

No

sponsoredwalk wrote: »

Yeah I mean there could be a sheep, or a lion or whatever behind the other
door, before I read this I really didn't think it was smart to switch
It is a mindfcuk alright but certainly makes probability look more interesting
as a subject.

Yeah, maybe the reason is because the question is phrased differently in different scenarios. I have seen the question phrased in such a way where you would be guaranteed to win the car if you switched, based on what the host says. Also, I don't think it is necessary at all to say that the Host knows where the car is (which they always seems to) as that is irrelevant to the question being put.

What's really crazy is how many mathematicians have disagreed with switching. One famous mathematician died apparently, still believing that there was no advantage in switching.

http://www.americanscientist.org/comments/comment_detail.aspx?id=10&pubID=3806

Added a Poll, let's see if there is anyone brave enough to say there is no advantage in switching

Another way of presenting the problem would be:

Three cards laying face down (one Ace & two Kings) and I ask you to find the Ace.

You choose card No.1 and I turn over card No.2 which is a King and say:

"You can choose again".

Is there an advantage in switching?

Yes

oeb

No

OutlawPete wrote: »

Three cards laying face down (one Ace & two Kings) and I ask you to find the Ace.

You choose card No.1 and I turn over card No.2 which is a King and say:

"You can choose again".

Is there an advantage in switching?

Yes

Only if you will always turn over a king. That is the key part of the monty hall problem. If you turn over a card at random, not being pre-informed yourself then the probability stays at a 1 in 3 chance. (This is why, for example, the monty hall problem does not apply to the likes of 'Deal or no deal')

OutlawPete

No

oeb wrote: »

Only if you will always turn over a king. That is the key part of the monty hall problem. If you turn over a card at random, not being pre-informed yourself then the probability stays at a 1 in 3 chance.

Sure it has to be a King (Goat) that is first turned over as otherwise the player would win, as the only other option is Ace (Car).

The first time I seen the 'problem' presented: it said that the host had made an error picked the wrong box to reveal a goat and did not know where the car was himself and then just gave the option to the player to choose again.

This is why I say I sometimes refer to it as a 'trick question'.

Anyway, uploaded an edit vid when the 'answer' is NOT revealed and so maybe see what people choose first time, that might not have being presented with the problem before.

oeb

No

OutlawPete wrote: »

Sure it has to be a King (Goat) that is first turned over as otherwise the player would win, as the only other option is Ace (Car).

The first time I seen the 'problem' presented: it said that the host had made an error picked the wrong box to reveal a goat and did not know where the car was himself and then just gave the option to the player to choose again.

This is why I say it can sometimes be a trick question.

Sorry to be pedantic, but if the host does not know where the car/ace/whatever is, then it is just not the Monty Hall problem.

OutlawPete

No

oeb wrote: »

Sorry to be pedantic, but if the host does not know where the car/ace/whatever is, then it is just not the Monty Hall problem.

I agree - that is my point.

It is the knowledge of the dealer (game show host) that makes it 'Monty Hall'.

If it was just a mistake to turn over the wrong card (open the wrong door) then it is a 1/2 chance that the player will reveal an Ace (Car) when faced with his final choice and so there would then be NO advantage in switching at that point.

Anyway, I'll edit my post to make it clear that the Host does know where the Goats are and it is a deliberate decision to open a door that contained a Goat.

take everything

Really interesting problem.
Takes a bit to get your head around.
I've seen the mathematical explanation but I'm just trying to get a better intuition for it.
Isn't it all to do with your random choice (1/3 chance of choosing a car) vs the host's non-random choice (deliberate 0 chance of choosing a car which skews things in your favour for the unchosen door).

That's just my general intuition for the problem, probably wrong somewhere.


Also: if it's explicitly stated the host knows (as opposed to just not said), do as many people get the answer wrong.
Wiki seems to say they do.
Which is interesting- because if it is explicitly stated, I wonder what the psychology is behind so many (like me) getting it wrong initially.

take everything

OutlawPete wrote: »

Hhhm, I guess I am presuming that if his initial guess is correct and he switches, he loses - but of course, that doesn't make it (mathematically) wrong to switch.

Real mindfcuk, but yeah - you're right

Yeah "initial correct, second guess wrong" is only the "unlucky" scenario out of the three possible scenarios left to you after the host does his thing.
The other scenarios of course being "initial wrong, second correct" and again "initial wrong, second correct".
So out of three scenarios the second guess would be correct twice (helped by the host's informed actions).

^This is my non-mathematician's understanding, open to correction/proper treatment.

Eliot Rosewater

No

I don't see how a very good mathematician could get this wrong. At worst you could compile a pedantic sample space.

Let the three possibilities be [LATEX]\displaystyle C, G_a, G_b[/LATEX] (i.e., the door you choose first has one of these three behind them). You have a one third chance of choosing one of these. We will consider each case separately.

Case 1 - you pick the door with [LATEX]\displaystyle C[/LATEX] behind it.

There are two possibilities - the host will open the door with [LATEX]\displaystyle G_a[/LATEX] (case 1.a) or the door with [LATEX]\displaystyle G_b[/LATEX] (case 1.b). Given that the probability of case 1 happening is 1/3, the probability of case 1.a happening is 1/6; ditto for case 1.b.

Case 1.a
To win the car you should not switch.
Case 1.b
To win the car you should not switch.

Case 2 - you pick the door with [LATEX]\displaystyle G_a[/LATEX] behind it.

The talk show host has only one choice - to open the door with [LATEX]\displaystyle G_b[/LATEX] inside. The door you have the opportunity of switching to has a car in it, so you should switch.

Case 3 - you pick the door with [LATEX]\displaystyle G_b[/LATEX] behind it.

The talk show host has only one choice - to open the door with [LATEX]\displaystyle G_a[/LATEX] inside. The door you have the opportunity of switching to has a car in it, so you should switch.

So in cases 1.a and 1.b you should not switch; in cases 2 and 3 you should.

The combined probability of cases 2 and 3 is 2/3, hence the probability that switching would result in a car is 2/3.

And obviously, the combined probability of cases 1.a and 1.b is 1/3, hence the probability that not switching would result in a car is 1/3.

Enkidu

Well, there are only three objects:
Donkey 1
Donkey 2
Car

Hence you can only pick three objects in you first choice.
If you pick Donkey 1, he shows you the door with Donkey 2 so switching gets you the Car.

If you pick Donkey 2, he shows you the door with Donkey 1 so switching gets you the Car.

If you pick the Car then switching will make you lose.

So in 2/3 situations switching is better.

The way I see it, opening the door explictly eliminates one of the options. If the host asked you if you wanted to switch without opening any door, it would be pointless, however he is explicitly eliminating one of the bad choices thus raising your chances.

RoundTower

a "very good mathematician" can get it "wrong" because the question is so often phrased ambiguously. Even the OP is still ambiguous: the host has to know what's behind all the doors and you have to know that he always chooses a door with a goat.

Fringe

I find it's more intuitive if you think of 100 doors with 99 goats and 1 car. Clearly, if you pick randomly, you'll have a 99/100 chance of picking a goat. Monty Hall knows what's behind each door and he now opens all the doors with the goats except for your one and the car. There's now two doors left. Since the door you picked is most likely a goat, the obvious choice is to switch.

OutlawPete

No

RoundTower wrote: »

a "very good mathematician" can get it "wrong" because the question is so often phrased ambiguously. Even the OP is still ambiguous: the host has to know what's behind all the doors and you have to know that he always chooses a door with a goat.

The reason I change my mind I feel is, as you eluded to: sometimes it is phrased incorrectly, as is the case here for instance:

http://www.askamathematician.com/?p=787

JamJamJamJam

No

I like how the poll is 100% in favour of changing.
You'd know we're in the maths forum

Fremen

No

Anyone have the guts to ask in After Hours?

Mellor

No

It's incredibly simple.
When you switch doors, you always change your prize. Seeing as you were twice as likely to pick a goat, you are twice as likely to get a car if you switch.
That's all ther is to it really.

OutlawPete wrote: »

Yeah, maybe the reason is because the question is phrased differently in different scenarios. I have seen the question phrased in such a way where you would be guaranteed to win the car if you switched, based on what the host says. Also, I don't think it is necessary at all to say that the Host knows where the car is (which they always seems to) as that is irrelevant to the question being put.

Maybe I spoke too soon :rolleyes:

There is no way of phrasing so that you are guaranteed to get the car (the fact that you could of picked the car first time around proves this)

The fact that the host knows where the car is is the single most important piece of info. If he opened a random door, he could of picked the car which ruins the whole thing. If this is the case then switching makes no difference.

In one of the original publishings, the fact that the host knew wasn't mentioned, which is why some of the matamaticians got it "wrong". Although in that case their was not right and wrong answer as it varied depending on if you assumed the host knew, or opened at random.

OutlawPete

No

Mellor wrote: »

Maybe I spoke too soon :rolleyes:

What's with the condescending roll-eyes?

If that's your bag, there's a thread in After Hours right now where a simple thanked post to post count ratio is eluding users. I'm sure you'll find loads of opportunities to smugly correct others there.

http://www.boards.ie/vbulletin/showthread.php?t=2056155756

I was referring to how this problem is phrased wrong in "different scenarios" (poorly worded perhaps) and that those are the different ways in which the question is presented incorrectly.

Mellor wrote: »

There is no way of phrasing so that you are guaranteed to get the car (the fact that you could of picked the car first time around proves this)

You would be surprised. I seen the promlem put where the person is told in the presented question (not by the host) that they car is behind one of the the other doors and also shown an goat and then asked would they switch.

Far and away though, it is far more popular online for the question to be put where the host does NOT know where the car is, than that he does know. Which, as I said above - is why I feel that there is still such debate.

Mellor wrote: »

The fact that the host knows where the car is is the single most important piece of info.

I agree, did you not see:

OutlawPete wrote: »

It is the knowledge of the dealer (game show host) that makes it 'Monty Hall'.

If it was just a mistake to turn over the wrong card (open the wrong door) then it is a 1/2 chance that the player will reveal an Ace (Car) when faced with his final choice and so there would then be NO advantage in switching at that point.

Fremen wrote: »

Anyone have the guts to ask in After Hours?

I see we have two votes who would not switch now, so I might pose the problem in a few days, the thread could get moved though.

Maybe I could edit the video in the OP to exclude the answer and present it like that, might be fun

Mellor

No

OutlawPete wrote: »

What's with the condescending roll-eyes?

If that's your bag, there's a thread in After Hours right now where a simple thanked post to post count ratio is eluding users. I'm sure you'll find loads of opportunities to smugly correct others there.

I wasn't trying to be smug.
I simply thought you were takign a simple subject, and making mistakes.
The roulette thread all over again.

I was referring to how this problem is phrased wrong in "different scenarios" (poorly worded perhaps) and that those are the different ways in which the question is presented incorrectly.

If its presented wrong, then it isn't monty hall.

You would be surprised. I seen the promlem put where the person is told in the presented question (not by the host) that they car is behind one of the the other doors and also shown an goat and then asked would they switch.

Where did you see this. Firstly, its not monty hall as i said.

Secondly, its retarded. You are told that the car is in a different door, ie you don't have it, and show where it isn't.
And asked would you switch.

How is that in anyway a problem.

Far and away though, it is far more popular online for the question to be put where the host does NOT know where the car is, than that he does know. Which, as I said above - is why I feel that there is still such debate.

No it isn't. If the host does not know, then it isn't monty hall. Simple as.

I agree, did you not see:

I did, but it completely contradicted your previous comments where you said it doesn't matter. Only when Oed point this out did you change your view.
I was just backing up Oed's post (I was aware that you'd already seen and replied to him)

I see we have two votes who would not switch now, so I might pose the problem in a few days, the thread could get moved though.

Not everyone understands or gets it, hence its a problem. The average person prob picks the wrong answer the first time (its so well known now that switching is clearly running away with it)

AH is full of people trolling, it would desend to nonsense pretty quick.

OutlawPete

No

Mellor wrote: »

I wasn't trying to be smug.

Using roll-eyes when you feel someone is wrong or is making an error, is being smug and condescending, not matter what you may think.

Mellor wrote: »

I did, but it completely contradicted your previous comments where you said it doesn't matter.

As I said, poorly worded - but that whole paragraph was trying to point the way others present the problem online, which is why it started with:

Yeah, maybe the reason is because the question is phrased differently in different scenarios.

Mellor wrote: »

If its presented wrong, then it isn't monty hall.

Mellor wrote: »

No it isn't. If the host does not know, then it isn't monty hall. Simple as.

You are not listening, I know it's not 'Monty Hall' when presented that way.

As I said:

OutlawPete wrote: »

It is the knowledge of the dealer (game show host) that makes it 'Monty Hall'.

Obviously if the above is not emphasised, then it is not 'Monty Hall' - but the fact of the matter is that the whole confusion surrounding the problem is because it has been phrased incorrectly time and time again and presented as 'Monty Hall'.

Saying: but then it's not Monty Hall is just stating what I have already said, for the sake of it.

tbh

OK, OP (works on so many levels) has requested we close the thread so he can start a more casual one elsewhere - i think that's a fair enough request.