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The Monty Hall Paradox





Afaceinthematrix
Here is a fun problem, that I love driving people insane with. Most people do not get the correct answer, and when I tell them the correct answer they refuse to accept it, all though occasionally someone will understand it. It's quite famous, so you may have heard of it. Let's see if anyone knows the answer...

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 other 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 take the switch?
Bockman
Not to ruin it right away, I'd say Marilyn Vos Savant would say yes, and I would disagree with her Wink

Be Well Cool
jharsika
I don't think so? Is the answer not really?
tidruG
This is an interesting question. I don't see how either changing or not changing could be to your advantage, since the odds remain the same.
Afaceinthematrix
tidruG wrote:
This is an interesting question. I don't see how either changing or not changing could be to your advantage, since the odds remain the same.


I always like to ask people this question, because I've always wanted to see someone get it correct right away. But I have never seen someone get it correct right away. But that answer is incorrect, the odds do not stay the same. I'll post the correct answer with the explanation in white font so that you have to highlight it to get the correct answer. That way people can still try and figure it out.

That's incorrect. That's what most people say at first, and that is where the arguing and shouting matches come in... Your chances are not 50/50, and very few people will understand this at first, but it has been mathematically proven that if you decided to switch you will have a 2/3 chance of winning. I was checking youtube and found 100's of videos explaining this and found hundreds of websites explaining this with a quick google search. But this is the explanation that I have always used when trying to explain this:

What the host of the game show is doing here is keeping the door you chose and keeping one other door that it could possibly be. So let's do the same thing but with 100 doors, because it is easier to understand it with the bigger picture.

Imagine you have 100 doors and 1 has a prize behind it and the rest have nothing. Now if you choose one door, your chances of winning are 1/100. Now let's divide these doors into two groups: the first group is just the door that you selected and the second group is the other 99 doors. Now there is a 99% chance that the door is in the bigger group. Now as the game show host, I know which door has the prize behind it. So what I'm going to do is eliminate every door except your door and the door with the prize behind it (unless you luckily chose that door, then I'll just eliminate every door except the one you chose and one other). So when I eliminate 98/99 doors in the second group, you will be left with two groups with one door in them each. Now the door that you didn't choose will probably have the prize behind it. The only way that it cannot have the prize behind it is if you chose the correct door the first time. But that only happens 1/100 times, so 99/100 times the door you didn't choose will have the prize behind it. Therefore you should switch doors. Because remember, there was only a 1% chance that your original guess was correct, and the prize door was most likely going to be in the group that I was eliminating from.

So the original problem has the same concept, except that it is harder to see because the numbers are much smaller. You have a 1/3 chance of getting your first guess correct, which means there is a 2/3 chance it will be in the second pile (with the other two doors).

This problem is a little tricky and it takes some people a long time to understand it.
tidruG
Ah, ok, your explanation makes perfect sense.
I'll put some more thoughts about this in a quote box, because my post will be on those light blue backgrounds, and white text would be easy enough to read on that.

Quote:
Once you explain it with 100 doors, it makes sense. However, in this particular case, there's a very high chance (comparatively) (i.e. 33%) that your original choice is correct. At the same time, if you are to choose one door out of 3 which you believe has a car behind it, you'd be looking for something like a gut instinct. In this case, since your gut instinct has a 33% chance of succeeding, you'd tend to not want to change it.

However, your basic explanation/reasoning is good, and I accept it Smile
hack_man_
I love this puzzle!

it makes a lot more sense with 100 doors, and that is the only way to get my friends to understand it! But with only 3 doors, it still applies, there is just less chance of it working...
Bockman
I would like to add something else to this, and to not ruin the game for others who may feel like answering it first, here it is in white over white background.

Quote:
I would just like to stress out that (from what I've read on this) it has NOT been mathematically proven. This hint was sent to a magazine who launched it in a and it raised havoc amongst the mathematical community in the U.S. And according to experiences made nation wide, the answer would be to switch, yes (again, I (Bockman) would not switch, but that's me. I trust my instinct). Still, there is no mathematical proof of this and no "equation" on this probability has ever been made (therefor, mathematically unproven)

When I mentioned Marilyn Vos Savant, it was a hint. She's the columnist that had this situation start. I may also add that she has one of the highest I.Q.'s according to tests. You may see more about this in marilynvossavant.com/articles/gameshow.html and en.wikipedia.org/wiki/Marilyn_vos_Savant (removed http: // to avoid the link turning blue)




Be Well Cool[/quote]
funkymunky
vvvvvveeerrrryyyyy confusing. It's a trick question?
Agent ME
The chance of the particular door does not change relative to the other doors when he opens the 3rd door.

When you start, all the doors have a 33.3% chance of being correct.
You pick one, so you have a 33.3% chance of being right.

If he shows you that one of the doors was wrong, and gives you a chance to reconsider, the two doors now both have a 50% chance of being right. The door you picked first has a 50% chance.
That doesn't make sense that the chance of your picked door would stay at 33% and the the 33% chance from the door he opened would be added to the second door neither of you picked. It's split between the remaining doors.

[200th post]
Nameless
A very simple way of demonstrating why switching is statistically better:

Each of the three doors can be winning, and each of the three doors can be chosen. This creates nine equally like outcomes before switching. Here's a quick look at each:

Winning door is A, you choose A - If you don't switch, you win. If you do switch, you lose.
Winning door is B, you choose A - If you don't switch, you lose. C is revealed as losing, thus if you switch to B you win.
Winning door is C, you choose A - If you don't switch, you lose. B is revealed as losing, thus if you switch to C you win.
Winning door is A, you choose B - If you don't switch, you lose. C is revealed as losing, thus if you switch to A you win.
Winning door is B, you choose B - If you don't switch, you win. If you do switch, you lose.
Winning door is C, you choose B - If you don't switch, you lose. A is revealed as losing, thus if you switch to C you win.
Winning door is A, you choose C - If you don't switch, you lose. B is revealed as losing, thus if you switch to A you win.
Winning door is B, you choose C - If you don't switch, you lose. A is revealed as losing, thus if you switch to B you win.
Winning door is C, you choose C - If you don't switch, you win. If you do switch, you lose.

Count them up. If you don't switch, you'll win 3 out of 9 games. If you do switch, you'll win 6 out of 9 games. Therefore, switching is the better option. Smile
Afaceinthematrix
Bockman wrote:
I would like to add something else to this, and to not ruin the game for others who may feel like answering it first, here it is in white over white background.

Quote:
I would just like to stress out that (from what I've read on this) it has NOT been mathematically proven. This hint was sent to a magazine who launched it in a and it raised havoc amongst the mathematical community in the U.S. And according to experiences made nation wide, the answer would be to switch, yes (again, I (Bockman) would not switch, but that's me. I trust my instinct). Still, there is no mathematical proof of this and no "equation" on this probability has ever been made (therefor, mathematically unproven)

When I mentioned Marilyn Vos Savant, it was a hint. She's the columnist that had this situation start. I may also add that she has one of the highest I.Q.'s according to tests. You may see more about this in marilynvossavant.com/articles/gameshow.html and en.wikipedia.org/wiki/Marilyn_vos_Savant (removed http: // to avoid the link turning blue)




Be Well 8-)
[/quote]

Okay, so maybe you wouldn't switch with 3 doors, because the odds aren't really that off anyways. But wouldn't you switch with 100 doors, where you are 99 times more likely to win by switching? Just try it.... have someone think of a number between 1-100 and then guess one and then have them give you another number and try either keeping your guess or what they tell you and you'll find that the majority of the time their number with be correct... it's the same concept just that the difference isn't as much.
DjinniFire
this one was on Numb3rs :]
one of the best television series in my opinion since if you pay attention you get some bit of learning through the show.

Best way to solve and understand is how nameless showed with every possible combination. I believe I saw a picture diagram for each combination which makes it really easy to understand.

Counter Intuitive stuff is just great :]
tidruG
Afaceinthematrix,
Like I said in my previous post as well, with only 3 doors, I wouldn't really change my option. But with a 100 doors, and armed with the knowledge that the host will eliminate incorrect doors, I will definitely switch.
hunnyhiteshseth
I would not like to get too involved in mathematics or probability of winning here (yes, i am very bad at probabilities Razz )

But logically making a switch is better, here is how:

In a game show generally host is there to entertain people, attract viewers & generate TRP. And more often than not, people love seeing other participant win as they try to relate with him. Thats why host generally leaves subtle hints for participants to tell them correct answer. (It happened a lot in "Who wants to be millionaire??" )

Considering above fact & host voluntarily opened the 3rd door, so it is a kind of subtle hint that you were making a wrong choice & should switch to next door.
Afaceinthematrix
hunnyhiteshseth wrote:
I would not like to get too involved in mathematics or probability of winning here (yes, i am very bad at probabilities :P )

But logically making a switch is better, here is how:

In a game show generally host is there to entertain people, attract viewers & generate TRP. And more often than not, people love seeing other participant win as they try to relate with him. Thats why host generally leaves subtle hints for participants to tell them correct answer. (It happened a lot in "Who wants to be millionaire??" )

Considering above fact & host voluntarily opened the 3rd door, so it is a kind of subtle hint that you were making a wrong choice & should switch to next door.


I don't agree with that philosophy because he's going to show you another door no matter what, rather you picked the correct one or not.
hunnyhiteshseth
Afaceinthematrix wrote:
hunnyhiteshseth wrote:
I would not like to get too involved in mathematics or probability of winning here (yes, i am very bad at probabilities Razz )

But logically making a switch is better, here is how:

In a game show generally host is there to entertain people, attract viewers & generate TRP. And more often than not, people love seeing other participant win as they try to relate with him. Thats why host generally leaves subtle hints for participants to tell them correct answer. (It happened a lot in "Who wants to be millionaire??" )

Considering above fact & host voluntarily opened the 3rd door, so it is a kind of subtle hint that you were making a wrong choice & should switch to next door.


I don't agree with that philosophy because he's going to show you another door no matter what, rather you picked the correct one or not.


If the host is always going to show next door, then it will depend on the way he is going to say "Do you want to switch doors?" The tone will do the magic here & give the solution.
Vrythramax
Isn't this topic better served in either the games section or even the philosophy forum?

Since high ranking member(s) have posted here without suitably moving the topic....neither will I.

But I believe this topic belongs elsewhere.
skygaia
This is an interesting question but little bit difficult for me.. Laughing
qscomputing
It's not really a paradox, more a simple probability problem with an unexpected result. Of course you should switch, for the reasons given by previous posters.
Davidgr1200
Anybody tried an experiment? Should be quite easy to do. You need to be two people of course. Three places to hide "the prize" and only one person knows where. repeat the following a (large) number of times.

Person A hides the object in one of the three places.
Person B makes his guess.
Person A shows one of the places which does not have the object.
Check which pace does have the object.

The object should then be in the third place about 2/3 of the time.
Do it 100 times and it should be in the third place 60-70 times.

Let's hear some experimental results.
friuser
this question is a little retarded. statistically it is not better. Mathematicians will use this quasi math to justify an explaination but the odds do not change at 3 doors. It changes with 100 doors but those are two different scenarios.

If you read any statistics book you will find a similar questions but explained through an elaborate mathematical solution. Don't fall for this type of reasoning because it is irrelevent to an existing question. It's academic.
Afaceinthematrix
qscomputing wrote:
It's not really a paradox, more a simple probability problem with an unexpected result. Of course you should switch, for the reasons given by previous posters.


I know it's not really a paradox, I just like to call it that because that's one of the names I've heard people call it.

From Wikipedia:
Quote:

The problem is also called the Monty Hall paradox; it is a veridical paradox in the sense that the solution is counterintuitive.
m-productions
@Afaceinthematrix
Quote:
This was a rather interesting thing to read about, and explaining it with the 100 doors thing I think made it super easy to understand.


@Bockman
Quote:
I did not take the time to read the links you posted about the other person (so if something I say is very much proven wrong in that link sorry =( ) But I think it has been proven that it is mathmaticly good to pick the other door (in cases where there are a lot of doors [no so much in case of 3]) as his post kind of proved it... so I dont seem to understand why you say it has not been proven. (this is more of a question than a statement xD)

That is unless of course you are referring to just the part with the 3 doors and not the part with the 100 doors....


@Vrythramax
Umm... this is the "general" chat area... this is kinda general... it dosnt just have to do with games, or math ...ect. A mix of them.. at least thats what I think of it ^_~
Davidgr1200
Anybody done the experiment yet?

Sorry Friuser, I don't understand what you are saying.
Do you mean that there is one explanation for 3.. up to 99 doors, and then another explanation for 100 doors and up? That seems even stranger , but you do not give any of the explanations, so perhaps you haven't understood the question?
Kovsieassassins
A friend tried to convince me not to change with the folowing reasoning

As the one group already has a door with a goat it is clear that the chance of having another one is greater.

um....
tidruG
Quote:
As the one group already has a door with a goat it is clear that the chance of having another one is greater.
No, that doesn't make sense to me at all Razz
There doesn't appear to be any logic behind that statement.
Kaisonic
I think Nameless up there proved it straight away with his answer alone. Of course, the first explanation (with the 100 doors) made plenty of sense to me at first anyway. The only thing is that a lot of people won't think of that stuff in the clutch (while they're playing the game) so they'll end up thinking they still have a 50/50 shot and probably lose anyway. Only the really smart and calm people could probably win this game in an actual game show.
Bockman
Kaisonic wrote:
I think Nameless up there proved it straight away with his answer alone. Of course, the first explanation (with the 100 doors) made plenty of sense to me at first anyway. The only thing is that a lot of people won't think of that stuff in the clutch (while they're playing the game) so they'll end up thinking they still have a 50/50 shot and probably lose anyway. Only the really smart and calm people could probably win this game in an actual game show.


Actually, only the only people that are perceptive would win this game. it has nothing to do with probabilities, it has to do with the hints the game show host gives.

Bottom line is.. in the end you have two doors (the one you picked and the one left closed you didn't pick) and one goat. You may very well change doors and loose. It all depends on the hints the guy gives you as he's opening the other doors.
It's implicit in the reasoning that the game show host ALWAYS opens all the doors but one from the doors you didn't pick. This means that in the end there will ALWAYS be 2 doors and one goat (thinking about it, it is really stupid to have the three doors when you will ALWAYS open one of them), giving the contestant a 50/50 chance.

As I said in my first post in the topic, I disagree with the woman that brought this game to life on the magazine, and I state that the probabilities are 50/50.

Be Well Cool
rvec
Something is wrong with this answer. There is no way you can rely on the host. The host will just try to get you to doubt. It makes the game more exiting to look at. And the show doesn't mind if you win or not, it has to be exiting and they need a couple of winners to let people know it's fair. If you choose the good one the host will try to get you to choose another one and if you have the wrong door he will do the same.

I am going to do some math on this. I will post it here when I am done.

Edit:
Code:
I you do NOT listen to the host (6/12 wins):
You chose   host opens     answer         win?
A            B            A            yes
A            C            A            yes
A            B            C            no
A            C            B            no

B            A            B            yes
B            C            B            yes
B            A            C            no
B            C            A            no

C            A            C            yes
C            B            C            yes
C            A            B            no
C            B            A            no

If you DO listen to the host (6/12 wins):

You chose   host opens      answer         win?
A            B            A            no
A            C            A            no
A            B            C            yes
A            C            B            yes

B            A            B            no
B            C            B            no
B            A            C            yes
B            C            A            yes

C            A            C            no
C            B            C            no
C            A            B            yes
C            B            A            yes   


If the host removes one possibility there are only 2 left, which makes the chances of winning 1/2. There is no way you can get any more info from what the host does.
erlendhg
That one was really tricky, actually.
Some might say as revealed first, that since it is most likely that the correct door is in the majority (the doors you did not choose) before the host opens a door, the probability to win would be greater if you switched doors.

Then some will say that since you ended up with two doors, the chanced are 50/50 - one is right and the other one is wrong.

I really can't decide what theory to like the most. For me, both are reasonable in a way.

But since the first one says "do switch", and the other one says it is totally random what to choose, the best would, in any cases, be to switch doors Razz

(At first I thought you meant that you choose a door (number 1), and the host tricks you and open door number 3, saying to you that it is door number 1 Brick wall I got really confused Razz Luckily I understood it quite quick after.)
rvec
There is no difference between the 2 doors left. No way the host will give away any clues and if he does, how would you know the clue isn't meant the other way around? The host only wants to give you doubt to make it more exiting to watch the show. If the guy was your friend I would follow his directions but since I have no reason to trust him I won't.
friuser
actually the wikipedia article is very informative with this question. I suggest everyone read it. Sadly I needed it to understand where my misunderstanding was. Good thing I don't gamble...
Afaceinthematrix
rvec wrote:
There is no difference between the 2 doors left. No way the host will give away any clues and if he does, how would you know the clue isn't meant the other way around? The host only wants to give you doubt to make it more exiting to watch the show. If the guy was your friend I would follow his directions but since I have no reason to trust him I won't.


It has nothing to do with the host giving you clues, since they're going to open a door without the prize behind it anyways. It has to do with there being a greater chance that you didn't pick the correct door to begin with.
Nameless
rvec wrote:
I am going to do some math on this. I will post it here when I am done.


Your math is flawed. You're looking at the situation the wrong way around. The winning door is chosen BEFORE anything else takes place, not last. In your current demonstration you've listed four possible options depending on what you choose, but you've made the mistake of assuming these are equally likely.

To clarify, in order for your table to be fair, you'd need to add for each choice ...

Code:
You chose   host opens     answer      win?      PROBABILITY
A            B            A            yes         1/6
A            C            A            yes         1/6
A            B            C            no            1/3
A            C            B            no            1/3


Your table is otherwise rather misleading and implies that your INITIAL guess has a 50% chance (compare 'you choose' to 'answer') when this clearly not the case.See my previous table for a more accurate representation of the result. Or run a large number of trials yourself. Also - the host 'giving you hints' has nothing to do with the maths of it, and the two should not be considered together as they are by some people.

(My previous table for accurate reference - And I dare anybody to find flaw in this logic.)
Quote:
Winning door is A, you choose A - If you don't switch, you win. If you do switch, you lose.
Winning door is B, you choose A - If you don't switch, you lose. C is revealed as losing, thus if you switch to B you win.
Winning door is C, you choose A - If you don't switch, you lose. B is revealed as losing, thus if you switch to C you win.
Winning door is A, you choose B - If you don't switch, you lose. C is revealed as losing, thus if you switch to A you win.
Winning door is B, you choose B - If you don't switch, you win. If you do switch, you lose.
Winning door is C, you choose B - If you don't switch, you lose. A is revealed as losing, thus if you switch to C you win.
Winning door is A, you choose C - If you don't switch, you lose. B is revealed as losing, thus if you switch to A you win.
Winning door is B, you choose C - If you don't switch, you lose. A is revealed as losing, thus if you switch to B you win.
Winning door is C, you choose C - If you don't switch, you win. If you do switch, you lose.
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