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HelicopterOff Topics › Bet all you math and logic wizzes won't get this one
06-02-2004 07:44 PM  13 years agoPost 1
helicopter34

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New Jersey, exit 82

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Bet all you math and logic wizzes won't get this one, well that is, unless you already heard this one.

Here is the question:

You are on a game show. There are 3 doors to choose from. Behind one of the doors is a ferrari and the other two doors are stupid conselation prizes. The host of the show knows what door the ferrari is behind and he does not want you to win. The game works like this, you pick a door and the host of the show will open one of the remaining two doors (he does not want you to win so he will invariably open a door that does not have the car behind it). Now the host leaves you with a choice, stay with the door you originally selected or you can switch to the one remaining door that was not opened (at this time, the door you originally selected and one of the doors you didn't select are closed and you have to choose which one you want to open for your prize, and one of these two doors has the ferrari behind it). Do you stay with your original choice or do you switch to the other remaining door? What are you odds of winning with each choice?

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06-02-2004 07:48 PM  13 years agoPost 2
Doug

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Port Saint Luice Florida....

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Stick with the original choice as the host would not offer you a change that would win you the car.

First member of Member of Bearings Anonymous

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06-02-2004 07:50 PM  13 years agoPost 3
Jeff H

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Cincinnati, OH

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Seems pretty obvious to me.
The host doesn't want you to win.
You have to pick the door with the car in it to win.
If the host opens a door that you didn't pick and it doesn't have the car behind it then you have already picked the winning door. Stay with your choice.
If you didn't pick the winning door initially, the host that doesn't want you to win will open the door with the car behind it.


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06-02-2004 08:44 PM  13 years agoPost 4
Phil Cole

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Menlo Park CA

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The best thing is to change to the remaining door after choosing one and being shown the consolation prize behind the door the host opens.

This way there is a 2/3 chance you will walk away with the car.

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06-02-2004 08:55 PM  13 years agoPost 5
helicopter34

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Maybe I didn't make the directions clear enough

Good job Phil, or have you heard this one already.

Alright let me try to reexplain the contest.

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06-02-2004 09:04 PM  13 years agoPost 6
noname

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Change door !

You start with a 1/3 chance of picking the car door and a 2/3 chance of missing it. After the host picks out one of the wrong doors, if you swich doors your probability of winning increases to 2/3, unlike if you stay with the original door where your chance of winning is 1/3.

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06-02-2004 09:04 PM  13 years agoPost 7
Phil Cole

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Menlo Park CA

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If I had already seen it I would not have spoiled the fun.

I have a BSc in mathematics, so I guess that I count as a math wizz.

You didn't specify the stakes, but you can buy me a beer if we ever meet.

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06-02-2004 09:12 PM  13 years agoPost 8
Davo

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You didn't specify the stakes
Well done and congratulations Phil! Your prize for being the maths genius is a genuine, brand spanking new (free) Ferrari (sticker set)!!!!!

Just go to http://www.shell.com/home/ferrari-e...p_stickers.html and fill out your details and you'll receive your very own Ferrari (sticker set)

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06-02-2004 09:14 PM  13 years agoPost 9
Doug

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Port Saint Luice Florida....

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I would disagree with you logic, as once one door is out of the game the odds are 50-50. Previous actions have no affect on future odds.

First member of Member of Bearings Anonymous

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06-02-2004 09:21 PM  13 years agoPost 10
helicopter34

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Maybe I didn't make the directions clear enough

Good job Phil, or have you heard this one already.

Alright let me try to reexplain the contest.

You pick your door. There are two doors left now that you didn't pick. Out of the two remaining doors that you didn't pick, the host opens one and removes it from the game (that door never has the car behind it). Now there are two closed doors left, one is the door you originally picked and the other is the door that was left after removing one of the doors that didn't have the car behind it. Now either the door you originally picked has the car behind it or the other door. Do you stick with your original choice or do you switch to the other door.

Answer:
Door [1] [2] [3]
Well from the beginning, each door has an equal probability of having the car. Therefore if you were going to pick a door and then stick with it no matter what, you would have a 1/3 chance of winning. But if you pick a door, there is a combined chance of 2/3 that the car is in one of the other two doors. So think about it like this, since the host removes one of the remaining two doors without the car in it, the situation basically is equivalent to saying that the host is allowing you to have car if it is in either of the remaining two doors if you decide to switch. Its almost like allowing you to pick both of the remaining two doors if you decide to switch.

I like this puzzle because its weird and it shows that even though the finial decision is between two items (you would think it is 50:50), there is not an equal probability between the two choices. Even from a pure probabilistic stance, the previous history of where the choices came from comes into play. Haha, I don't even have a minor in math (but I did take calc 5 ), so some of these terms that I used I might have made up, but bear with me. The fallacy is like saying it will either rain or it will not rain so theferfore there is a 50:50 chance. We know that it doesn't rain half the time, so there must be more to the equation.

Probability that we are used to is the type that you almost have to assume there are equal probability between choices. For instance if you are throwing rocks onto a 4x4 grid with equal area blocks and a rock has to land somewhere on the grid, the total possibilities are 16 so we have to assume that each of the possibilities are equiprobably in order to say that there is a 1/16 chance of landing on each spot. That is the basis probability. But what if your style of throw favors you to land in a particular range of squares. In this case the probabily between the possibilites is not equal so you can't just use pure probability alone. Its almost like circular logic that you can only use probability if there is equal probability....

Something weird that I like think about is that if you were were to try to predict the probability (not from a probability standpoint) of something occuring at a future instance, and if you were to incorporate every single possible variable into your analysis of the probability, the more variables you include, the closer you would get to 0% chance or 100% chance because in that future instance, what you are deriving the probability for can either happen or it won't happen. This is to say that you believe in determis, which I sorta do, and you don't believe in anything from quantum mechanics or natural randomness.

And to get completely off topic and talk about determism
If you believe in physics, you know that everything has a prescribed way of happening (whether we have the capabilities to understand the governing equations or the ability to account for the vast number of variables), and given a large enough set of initial conditions, there can be only answer to what will happen.

Well what about quantum mechanics, and the seemingly random occurance of stuff. Well lets say that things are random at the quantum level... Do you believe in free will?? I do, but if you think about it like this (what im about to say) then it becomes confusing.

At any given point in time, there are a vast number of conditions that occur. At the next instant (bear with me, whether you believe in quantum time units or not, you can get the gyst of what I'm trying to say) what happens has to be determined by the
1.) conditions from the previous instant and
2.) random stuff the goes on
So from a persons point of view, he only "has" control of those conditions that were passed over from the previous instant. Well if you make an infinite regression you back in time from instant to instant, you can see that the only control of person has over his whole life is the conditions that occured at the "beggining of time", which when you think about it, a person doesn't even have control of that (he wasn't even born yet). So nothing that happens to or will happen to a person is in any way controlled by the person's free will, but rather enivitability.

Well, if you followed that, you can see how its confusing. Its like a mathematical proof, sometimes in a math proof you can follow each step with clarity and each step is justified but what all the steps lead to is an end which is not at all clear and hard to accept (hence the usefulness of math proofs). So are we suppose to look at this proof and say well I have no control over anything and lose our grip on reality. No, its not something you can or should really accept other than just being interesting to note.

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06-02-2004 09:54 PM  13 years agoPost 11
ScareCrow_Delta

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Sebastian, FL

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uh......hmmm.......uuhh... okey?


~~~~ Defy the laws of gravity....gracefully ~~~~

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06-02-2004 10:03 PM  13 years agoPost 12
spaceman spiff

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Tucson

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Oh sh!t! i'm out of aspirin!

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06-02-2004 10:03 PM  13 years agoPost 13
Davo

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Its almost like circular logic that you can only use probability if there is equal probability....
Hmmm, not sure I agree with this statement. I can say that the probability of a fatal plane crash is 1 per 1,000,000 hours, whilst a fatal car crash is 1 in 10,000 hours. Just cos they don't have equal probabilities of catastrophe doesn't mean I can't use a probabalistic approach to determine which mode of transport to use....

Or am I missing something here?
were to try to predict the probability (not from a probability standpoint) of something occuring at a future instance, and if you were to incorporate every single possible variable into your analysis of the probability, the more variables you include, the closer you would get to 0% chance or 100% chance
This is where chaos theory comes in right? Infinitesimally small changes in initial conditions can have drastic effects on the output, therefore it is impossible to predict the outcome of a complex system accurately. Also Heisenberg's Uncertainty Principle says you can't measure anything accurately cos you'll change it in the process.

As for the free will arguments, I don't know what you're on about and I don't want to comtemplate it or I might lose the will to live

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06-02-2004 10:46 PM  13 years agoPost 14
spaceman spiff

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Tucson

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A man flips a coin 3 times and it comes up heads.

What are the odds that it will come up heads again on the next flip?

50% of course. The previous flips have nothing to do with those odds.

After the empty door is exposed, the odds go from 1/3 to 1/2 but there is no justification to change choice. The odds are still 1/2 once the choice is between two.

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06-02-2004 11:08 PM  13 years agoPost 15
nheather

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Horsham, West Sussex, UK

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Here about the regular airline passenger who always took a bomb with him when he travelled.

He had read that there is a small chance that a plane may have one bomb on board, but the probability of two is virtually zero!!!!

Cheers,

Nigel

p.s. If the show host always opens a door after the contestent has made his choice then I agree with Spaceman Spiff

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06-02-2004 11:33 PM  13 years agoPost 16
Torkroll

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Bakersfield Ca

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no, odds are 50/50, you are picking between 2 doors.

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06-02-2004 11:39 PM  13 years agoPost 17
Doug

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Port Saint Luice Florida....

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50-50 anything else is "new math"

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06-02-2004 11:43 PM  13 years agoPost 18
Phil Cole

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Menlo Park CA

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You guys are forgetting that the host chooses a door that he knows doesn't have the prize behind it. So, the door he chooses depends on the door that the contestant chose.

Let's say that the contestant decides beforehand not to change his mind.

He chooses a door. The probability that the car is behind it is 1/3.

If the car is truely behind the chosen door, the host can open any of the other two doors to reveal a pile of junk. This does not affect the probability that the correct door was originally chosen, so staying with the original decision gives a 1/3 chance of winning.

If the original choice was not the door hiding the car, then the host has only one door to open if he doesn't want to reveal the prize.

So, if the original choice was wrong, and another consolation door is opened, the car is behind the remaining door.

The probability of choosing a consolation door on the first choice is 2/3. I.e. it's more likely that a wrong door will be chosen first up.

If a wrong door is chosen then the car will always be behind the remaining door.

The "change your mind" plan will always get the prize if the initial choice is a consolation prize door. If the initial choice is correct, then the "change your mind" plan will fail. But there is a 2/3 probability that the initial choice is bad, so the "change your mind" plan is the way to go.

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06-02-2004 11:46 PM  13 years agoPost 19
nipps85

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gurnee illinois

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I did this problem when I taught a beginner statistics class at a university. It was from a game show... let's make a deal... I don't remember. It's very counter-intuitive and almost impossible to explain by typing it out, so I won't try, but think about it hard enough and it will be clear that it's not a fifty-fifty probability for the last two doors.

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06-02-2004 11:48 PM  13 years agoPost 20
hingus2000

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its 1/2. as said, basically, the fact that there was 3 doors in the first place is just a distraction. there was a 1/3 chance at first. a wrong door is taken away, we now have 2 doors, 1 correct. 1/2 chance!!!

Tom

T-Rex SA, Anodized Head, Gy240, 3s TP 2100mah, Align 35A ESC, 430L Motor

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HelicopterOff Topics › Bet all you math and logic wizzes won't get this one
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