#231433 - 24/08/2004 11:01
Car and goat conundrum
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old hand
Registered: 14/08/2001
Posts: 886
Loc: London, UK
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Imagine you are on a game show and you have reached the final. You are in with the chance of winning a brand new car (something [censored] like a Corsa)
There are 3 doors. Behind one of them is the car, but behind each of the other two is a goat.
You pick a door, and to give a bit of dramatic suspense, the game show host opens one of the other doors (he knows which one the car is behind) to reveal a goat.
He then offers you the chance to change your mind and pick the other unopened door, or stick with your original choice.
What should you do to maximise your chances of winning the car?
a) stick with your origninal choice
b) change your mind and go with the other door
c) it doesn't make any difference. The odds are 50:50
What is the correct answer, and detail as simply as possible why.
_________________________
Mk2a RioCar 120Gb - now sold to the owner of my old car
Rio Karma - now on ebay...
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#231434 - 24/08/2004 11:06
Re: Car and goat conundrum
[Re: furtive]
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carpal tunnel
Registered: 29/08/2000
Posts: 14496
Loc: Canada
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Old story. Switch doors and double your odds.
-ml
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#231435 - 24/08/2004 11:08
Re: Car and goat conundrum
[Re: furtive]
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carpal tunnel
Registered: 18/01/2000
Posts: 5683
Loc: London, UK
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Quote:
There are 3 doors...
Spoiler: the correct answer is here.
_________________________
-- roger
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#231436 - 24/08/2004 11:10
Re: Car and goat conundrum
[Re: Roger]
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old hand
Registered: 14/08/2001
Posts: 886
Loc: London, UK
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I know the correct answer. What I want is a simple explanation to non-believers
_________________________
Mk2a RioCar 120Gb - now sold to the owner of my old car
Rio Karma - now on ebay...
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#231437 - 24/08/2004 11:37
Re: Car and goat conundrum
[Re: furtive]
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carpal tunnel
Registered: 18/01/2000
Posts: 5683
Loc: London, UK
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Quote:
I know the correct answer. What I want is a simple explanation to non-believers
Then I suggest they read that link. There's 3 or 4 different explanations, varying in simplicity.
_________________________
-- roger
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#231438 - 24/08/2004 12:29
Re: Car and goat conundrum
[Re: furtive]
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carpal tunnel
Registered: 25/12/2000
Posts: 16706
Loc: Raleigh, NC US
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I don't think that any of those explanations are very good. Consider this table of outcomes:
Code:
x x +
-----
o x +
x o +
x x o
The first line represents the starting, pre-intial-choice position of goats and car, where 'x's represent goats and the '+' represents the car. Underneath are the choices made. The obvious next step is what door Monty opens, and the obvious solution to this is:
Code:
o X +
X o +
X x o
Which shows us that it would be positive for us to switch, which seems counterintuitive. However, this is not really correct. The table should be:
Code:
o X +
X o +
X x o
x X o
Which shows us that switching makes no difference.
The crux of the apparent paradox is that Monty knows which doors hide goats, which means that the last two possibilities collapse into one (like the "incorrect" table), since it is irrelevant to Monty which of those two doors he opens. The problem is that it is not random, despite being presented as such.
_________________________
Bitt Faulk
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#231439 - 24/08/2004 15:40
Re: Car and goat conundrum
[Re: furtive]
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carpal tunnel
Registered: 20/12/1999
Posts: 31604
Loc: Seattle, WA
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Quote:
I know the correct answer. What I want is a simple explanation to non-believers
I went through this myself a few weeks ago... trying to understand why switching makes a difference, and it took me a bit of thinking to come up with a layman's explanation.
Here is my layman's explanation.
1. Because you chose first, and...
2. Because Monty always knows which door has the car, and...
3. Because Monty always has to show you a goat...
Then your choice limits his choice. His choice is non-random most of the time. In fact...
If you chose a goat to begin with, then he has no choice at all. He's got to show you the other goat.
And since your chance of choosing a goat to begin with was 2 out of 3, then 2 out of 3 times, Monty is showing you the other goat.
So 2 out of 3 times, if you switch, you win.
Yes, it's totally counterintuitive, but it works only because of the three conditions above. If you didn't choose first or if Monty didn't know where the car was, then it would be random.
Here is another explanation with a roulette-wheel metaphor showing why Monty's knowledge changes the outcome.
Here are two simulators which can act as proofs:
http://www.grand-illusions.com/simulator/montysim.htm
http://math.ucsd.edu/~crypto/Monty/monty.html
Enjoy!
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#231440 - 24/08/2004 18:04
Re: Car and goat conundrum
[Re: tfabris]
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old hand
Registered: 28/12/2001
Posts: 868
Loc: Los Angeles
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We were discussing this over lunch, and the best way we found was just to walk through it. You have 3 doors, A, B and C and the prize is behind C.
If you choose A, he reveals B, and the correct choice is to switch to C.
If you choose B, he reveals A, and the correct choice is to switch to C.
If you choose C, he reveals A or B, and the correct choice is not to switch from C.
So in two out of three original choices, it is better to switch.
_________________________
Ninti - MK IIa 60GB Smoke, 30GB, 10GB
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#231441 - 24/08/2004 19:10
Re: Car and goat conundrum
[Re: ninti]
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carpal tunnel
Registered: 20/12/1999
Posts: 31604
Loc: Seattle, WA
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#231442 - 24/08/2004 19:43
Re: Car and goat conundrum
[Re: ninti]
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addict
Registered: 25/06/2002
Posts: 456
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Quote:
So in two out of three original choices, it is better to switch.
Indeed. So in the language of probability (or oddsmakers), you have a 1/3 chance of winning the prize if you don't switch, and a 2/3 chance if you do switch. I.e., doubling your odds and bringing us full circle back to mlord's first response. 
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#231443 - 24/08/2004 20:03
Re: Car and goat conundrum
[Re: music]
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carpal tunnel
Registered: 20/12/1999
Posts: 31604
Loc: Seattle, WA
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Right. But when you hear it in those words, you don't believe it because it's so counterintuitive. What Furtive was looking for was a way to explain it to nonbelievers so that they would understand. Even after I'd read a few explanations, it still sounded wrong to me. I had to turn it over in my mind enough times to get the crux of the problem in the right place so that my puny brain could accept it as logical.
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#231444 - 24/08/2004 20:07
Re: Car and goat conundrum
[Re: music]
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veteran
Registered: 08/05/2000
Posts: 1429
Loc: San Francisco, CA
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Me 2. I can now undertstand it based on ninti's 3 scenarios, but I still don't understand why it's not 50/50. It feels like those scenarios are overlooking something...
- Jon
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#231445 - 24/08/2004 20:10
Re: Car and goat conundrum
[Re: jbauer]
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carpal tunnel
Registered: 20/12/1999
Posts: 31604
Loc: Seattle, WA
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Quote:
but I still don't understand why it's not 50/50.
Because your choice made Monty's choice non-random.
If you hadn't chosen first, then yes it would be 50/50. But you did choose first, so your choice forces Monty's choice 2 out of 3 times.
Reread my explanation and Ninti's explanation very carefully and think about each line. The "aha" is in there, you just have to think about it.
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#231446 - 24/08/2004 20:14
Re: Car and goat conundrum
[Re: jbauer]
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pooh-bah
Registered: 15/01/2002
Posts: 1866
Loc: Austin
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taken from the wikipedia page linked above. this lended a huge hand to my grasping the concept
It may be easier for the reader to appreciate the result by considering a hundred doors instead of just three. In this case there are 99 doors with goats behind them and 1 door with a prize. The contestant picks a door; 99 out of 100 times the contestant will pick a door with a goat. Monty then opens 98 of the other doors revealing 98 goats and offers the contestant the chance to switch to the other unopened door. On 99 out of 100 occasions the door the contestant can switch to will contain the prize as 99 out of 100 times the contestant first picked a door with a goat. At this point a rational contestant should always switch.
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#231447 - 24/08/2004 20:17
Re: Car and goat conundrum
[Re: jbauer]
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carpal tunnel
Registered: 20/12/1999
Posts: 31604
Loc: Seattle, WA
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Quote:
It feels like those scenarios are overlooking something...
Ninti's scenario doesn't overlook anything. Although it posits a scenario where the car is behind door C, you could replace all of the A B and C variables with nonspecific terms and it still works. Like so:
If you choose the first goat, he reveals the second goat, and the correct choice is to switch.
If you choose the second goat, he reveals the first goat, and the correct choice is to switch.
If you choose the car, he reveals one of the goats, and the correct choice is not to switch.
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#231448 - 24/08/2004 20:20
Re: Car and goat conundrum
[Re: RobotCaleb]
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carpal tunnel
Registered: 20/12/1999
Posts: 31604
Loc: Seattle, WA
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I'd read the 99/100 explanation, but it didn't do it for me. Because I wanted to say "but Monty isn't opening 98 doors, he's opening one door."
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#231449 - 24/08/2004 20:22
Re: Car and goat conundrum
[Re: tfabris]
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veteran
Registered: 08/05/2000
Posts: 1429
Loc: San Francisco, CA
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Yep, I see the 3 scenarios, and it makes sense that in 2 out of 3 cases, your win by switching. I guess it makes sense...
It's weird. Totally counter intuitive, but it makes sense.
- Jon
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#231450 - 24/08/2004 20:26
Re: Car and goat conundrum
[Re: tfabris]
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carpal tunnel
Registered: 23/08/2000
Posts: 3826
Loc: SLC, UT, USA
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Quote:
If you hadn't chosen first, then yes it would be 50/50. But you did choose first, so your choice forces Monty's choice 2 out of 3 times.
That made it click for me for some reason. It's the forcing of Monty's choice bit that's key.
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#231451 - 24/08/2004 20:30
Re: Car and goat conundrum
[Re: loren]
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carpal tunnel
Registered: 20/12/1999
Posts: 31604
Loc: Seattle, WA
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Quote:
That made it click for me for some reason. It's the forcing of Monty's choice bit that's key.
Yup.
Another way to re-word Ninti's explanation could be:
If you choose the first goat, he is forced to reveal the second goat, and the correct choice is to switch.
If you choose the second goat, he is forced to reveal the first goat, and the correct choice is to switch.
If you choose the car, he reveals either one of the goats, and the correct choice is not to switch.
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#231452 - 24/08/2004 20:49
Re: Car and goat conundrum
[Re: tfabris]
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carpal tunnel
Registered: 23/08/2000
Posts: 3826
Loc: SLC, UT, USA
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#231453 - 24/08/2004 20:58
Re: Car and goat conundrum
[Re: tfabris]
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carpal tunnel
Registered: 12/11/2001
Posts: 7738
Loc: Toronto, CANADA
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The twist is that you aren't picking 1 door if you switch. You are picking 2 doors because Monty has already eliminated one when you made your first pick. That's precisely the 2/3 odds. When you first look at it, you break it down in your mind as a different problem: There are only two doors. One of them has a prize. Which one do I pick. It's not like flipping a coin. That's why it "seems" counterintuitive. The 100 doors example does help because it makes the odds more obvious by using larger numbers (for me anyway - this is where it first clicked). 1/100 of guessing correctly. Or the other way, 99/100 to get it wrong. When 98 of the incorrect ones are eliminated, you know one of those two has the prize. Do you feel confident in that 1/100 shot? Well, better to feel confident in the 99/100 that you were wrong. Which means if you switch, you win. Basically you can look at it as using the odds of which does not contain the prize. Because the winning door will never be the one to be revealed. So if the odds are 2/3 you'll get it wrong, play those odds by switching. Of course this doesn't help if they're moving the car and goats around back stage. Then all bets are off.  Bruno
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#231454 - 24/08/2004 22:12
Re: Car and goat conundrum
[Re: tfabris]
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addict
Registered: 25/06/2002
Posts: 456
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Quote:
Right. But when you hear it in those words, you don't believe it because it's so counterintuitive. What Furtive was looking for was a way to explain it to nonbelievers so that they would understand.
Believe me, I can relate!
The first time I ran across this conundrum (almost exactly 10 years ago, when Marilyn vos Idiot ran it in her column) I didn't grok it and had to blow half an hour writing, running, and playing with a simulator in C in order to convince myself. (Hey, I figured a few hundred thousand trials would make it clear to me.) And what do you know, I won 66.66% of the time when I switched and 33.33% of the time when I didn't.
Of course, just the process of writing the code kind of clarified my thinking. Only then did I remember this little thing called "conditional probability" from a class in "Random Variables." D'oh.
However, now the solution has seemed so obvious to me for so long that I can't understand why it ever seemed counterintuitive. So, weirdly, I'm having a hard time wrapping my mind around exactly what little *aha* push someone else might find useful.
My point in my previous post was just that ninti's explanation seemed to clarify everything very succinctly and provide a good understanding to mlord's "double your odds" comment.
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#231456 - 24/08/2004 22:54
Re: Car and goat conundrum
[Re: tfabris]
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carpal tunnel
Registered: 14/01/2002
Posts: 2858
Loc: Atlanta, GA
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Quote:
I'd read the 99/100 explanation, but it didn't do it for me. Because I wanted to say "but Monty isn't opening 98 doors, he's opening one door."
But of course the point is he's opening all but one in both cases. Still I'll admit that explination didn't work as well for me as:Quote:
By switching, the player is ensuring that he will win if he originally picked a goat. The probability of picking a goat was 2/3, so the player should switch.
_________________________
-Jeff
Rome did not create a great empire by having meetings; they did it by killing all those who opposed them.
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