The 3 Doors problem, starring Nyet, vallor, gloriae, &c.

Let's play a game called "Chase the Lady" You need 3 playing cards, one of which is a Queen. You lay them on the table - the punter puts his money on a card. You turn one of the other cards and it's not the queen What are the chances that the punter picked the queen? Still 1 in 3. What are the chances that the other card is the queen - well they have to be 2 in 3. If you don't believe it get 3 cards and play Chase the Lady until you do. How hard is that to understand? Now read how the big brains of #mensa tackle the problem.

SirOracle a.k.a. the bonehead is Fermat, a French mathematician (born 1601, died 1665) who had nothing to do with this problem.

Session Start: Sat Jun 21 03:16:19 2003
* Now talking in #mensa
Fermat Suppose you're on a gameshow. The prize is a car, behind one of three doors. You choose one, The host opens one of the other doors, and shows you there's no car behind it. Then he says you can choose the other door, if you like. Does it make any difference if you do? If you can work out the right answer in five minutes, then you have a Mensa level I.Q.
Clampy It does.
Fermat If you can work out the answer in less than one minute, then you've met the problem before.
Non` it doesn't
Fermat Reasoning please. You have about four minutes.
Non` you have already made a choice and he has only confirmed that it isn't wrong by opening the door
beezy does it make any difference?
Non` not that the other has a better chance of being right
vallor if you present the problem to the channel like its your own, you're a complete bonehead
Clampy You're 2/3rds likely to not have picked the right door from the getgo.
Fermat It's a well know problem.
beezy first, youre rehashing a stupid marilyn vos savant question - but youre doing it wrong.
Non` am i right?
Non` I said it doesn't since you have a 1 in 3 chance either way
Fermat About three minues
Non` when he opened the other door it only confirmed that your door isn't wrong...
beezy oracle? Is that you?
Non` not that the other is right
* vallor ignores "Fermat" the bonehead
Fermat vallor doesn't know
beezy i think its oracle, the bonehead
Fermat two minutes
Clampy Non, it's only 1/3rd chance you picked right.
nyet picking between 2 doors is a 50/50, not a 1/3
Clampy There's a 2/3rds chance it's one of the other doors.
Non` hey Fermat did you know there is a sweet candy center to your eyeball? why don't you see how many scratches it takes to get to the center
vallor *sigh*
vallor it's the monty hall problem -- and savant was wrong. it doesn't matter
nyet the 2/3rds is irrelevant once the empty door is open clamps
Clampy So, if you find that it's not one of those. There's a 2/3rds change on the last door.
Fermat one minute
nyet after that you either chose right, or wrong. 50/50
Non` Clampy: you choose one of three doors... the host opens one that you didn't choose. which only confirms that you aren't wrong
Clampy nyet, well, at that point, your chances stay the same.
Non` it doesn't improve your chances to choose another door that isn't wrong
Clampy You're only 33% right from the start.
Fermat nobody?
nyet if doors are a, b, and c
nyet shut up fermat we're talkin
Non` Fermat you're an idiot
nyet and you choose a
Fermat time's up
nyet and he opens door b to show you there's no car
Non` thank you Clampy, that is what I said
dumbier hi peeps
* PythonS sets mode: +o nyet
Clampy Actually, your odds are better if you change doors.
nyet no they're not you're just tricked into thinking so
Non` no they aren't... unless you can tell me how you figured that
dumbier doughnuts!!
Clampy How so?
beezy oooh doughnuts?!
Non` i wasn't tricked,,,, you have the same chance
nyet ok, you have doors a b and c, you pick a, host opens b, no car behind b, a and c are closed
Non` which only confirms that the door you chose wasn't wrong
Clampy nyet, let's take an extreme example here.
nyet and he says 'you can choose again, or you can stay with door a'
Clampy Let's say there's 100 doors.
nyet either way, it's a 50/50 crap shoot at that point
Clampy 99 are empty, and the 1 left has something behind it.
* dumbier imagines a hundred doors
Fermat Let's say there are ten other doors and your host open nine of them and shows you there's no car. What are the chances for the other door then?
Clampy Now, if you picked a random one, you'd have a 1% chance to pick right. The host now opens up 98 doors, all empty.
vallor nyet: the odds for a _new_ choice are 50/50 -- odds for the old choice are still 1/3 -- it's a semantics trick, stupid at any even
Clampy Would you switch then?
vallor t
nyet and you have two doors left.
Non` well yes, he opens more than one door
Non` your odds improve
nyet vallor: but choosing to keep the same door is still a NEW CHOICE. so the semantics trick is bullshit too. because you *can't* change your choice to the door that's open, you either choose a or c
Non` your odds improve. but to change your choice doesn't improve your odds
Fermat Verdict - vallor, nyet, non - you'll never get it. Clampy, you already knew.
nyet just cause your second choice of a 1 out of 2 might be the *same decision* as your 1 out of 3
nyet fermat, shut up
Clampy nyet, it's very unlikely your first choice is correct. You're 99% likely to NOT pick the right door.
nyet doesn't mean you haven't made a new choice.
Non` Fermat, you are a moron who doesn't know the answer to his own question
vallor it's still a semantics issue, not a real problem -- yes, the second choice is 50/50
nyet clampy: i'm not talking about the first choice. I'm talking about the second choice
* beezy has quit IRC
nyet you have two doors left, one has something in it, one doesn't. Which do you choose? a or c It's a 1/2 chance whether you choose a or c. No matter what you chose before.
Clampy nyet, if you don't know which door was 'protected', it does turn into a 50/50 chance.
Non` Clampy: if the host keeps opening doors that you didn't choose your odds go way up that the door you did choose was right to begin with
vallor I take it you disagree with Fermat, the bonehead?
Clampy But if you do know which door was, then it's different.
nyet wtf do you mean 'protected'? You don't know You have 2 identical doors
Clampy nyet, he doesn't open the door you choose.
nyet he never opens the door you choose. That's the way the game works
Clampy Yeah, I know.
nyet it always gets down to 2 doors. 1 with the prize, and 1 without
Clampy You've got a 1% chance of success with your first choice, right?
Non` yes. 1 in 100 doors have the car
nyet you're ignoring the fact clampy, that in the end, if you've got door 1 which was your original choice left, and door 37 left. you are choosing between 2 doors
Non` then 1 in 99, but does changing to another door help you? that is the question
nyet it's not your 'original choice' anymore
Non` does changing doors help you and the answer is NO
Clampy nyet, it's a more complex problem than that.
Fermat Choose the other door. It's 2 to 1 in favour. The only way you can lose is by picking the right door first.
dumbier look, the first idiot has to choose wrong for the second idiot to win. so 2/3 ways the first idiot can screw up times 1/2 ways the second idiot can win is still ONE IN THREE, THE SAME AS THE FIRST idiot.
nyet clampy, you're flat out wrong. If you *couldn't* pick again, yes, the odds would be bad
Fermat Sorry, nyet. You're flat out dim.
nyet but the last decision you make, you're thinking of it wrong
Clampy Fermat, shaddup.
Non` nyet you and I are right. We know it and it is very difficult to explain over IRC
nyet because the host is asking 'do you stay with your original choice', but the *real* question is 'which of these 2 doors do you choose?'
Non` i guess Detneters just own Mensans
Fermat I will Clampy. I'll leave it to you.
Non` ---former detnetter
nyet they'd never ask it that way because it's less dramatic
Clampy nyet, the odds are better to switch.
dumbier ok does the hosting idiot open all the doors for some moronic reason?
nyet no they're not Clampy, because you're not 'switching'. You're making A CHOICE between two doors
Clampy Yes, you are.
Non` how are the odds better to switch????
nyet no, yer not
vallor huh?
Non` when all the doors are equal?
vallor odds are better to switch? says who?
nyet no matter how he phrases the question, what he's asking is 'which of these 2 doors do you want?'
Clampy But, given the little thing you do before you choose a door to open, it makes the other door more likely to have the prize.
nyet there is no 'switch'
Non` how are the odds better to switch???? when all the doors are equal?
Fermat This is the perfect accompaniment for this bottle of Chardonnay.
Clampy nyet, you could plot out all the scenarios possible.
nyet there are only 2 possible scenarios when you only have 2 doors. You choose right
Clampy In one scenario, you pick the right door, switch, and get it wrong.
nyet you choose wrong. NO because there is no switching, you make choice A, and then you make choice B
dumbier the odds for the first idiot and the second idiot are the same
Non` are you sure either of you two qualify for mensa?
nyet choice B is a separate and arbitrary choice.
vallor I remember hearing about idiot savants discussion of this problem -- what was her choice?
Non` you are making a very simple probability problem VERY complex
nyet first time he asks you to choose between 3 things
Clampy nyet, nah, your options left in choice B reflect your choice of door in choice A.
nyet then he says choose between 2
dumbier the thing is to never participate in or watch gameshows. that way you always win
vallor I suspect this "fermat" has Parade magazines enshrined on his walls
nyet clampy: it doesn't matter tho, because by the time you get to choice b, you know one door's got the car, and one's empty
dumbier what's that magazine? fun with a purpose?
Clampy nyet, it does matter, as long as you're still looking at the first group of doors you first saw.
nyet so whether i choose door 1, or door 2 at that point the odds are 50/50 on each door. Nope, it doesn't
Fermat You can figure out so much, vallor? What do you do for a living?
Non` ok ok ok.... HE IS NOT saying "do your odds improve as I reveal doors" we all know they do. You have a better chance of getting the right door when there are less doors. HE IS ASKING "does it make a difference to change doors" and it doesn't since all doors are equal.
vallor dumbier: that's the column where idiot von savant had her column -- no clue if she still does
dumbier is there a kid's illustrated mag?
Blofeld23 I'm afraid that when you have a choice between 2 doors, regardless if there used to be a third option or not, the odds are at that point 50/50.
Clampy But the odds are only 1/3rd correct that you pick right from the getgo, leaving odds of 2/3rds that it's behind one of the other.
dumbier lol
nyet clampy jesus
dumbier idiot von savant
nyet you need to fuckin forget the third, because it's NOT THERE
Clampy One door is removed as an option, so then the odds there are 2/3rds in your favor.
nyet at the point he asks you to choose between door 1 and door 2. It's NOT THERE You now have to choose between 2 doors
Clampy Only if you just come along at that point.
nyet wrong
Clampy If you have the knowledge of the past, the odds get better.
nyet no, it doesn't, not in this case
Clampy They do.
nyet it's like tossing a coin clampy
Non` "But the odds are only 1/3rd correct that you pick right from the getgo, leaving odds of 2/3rds that it's behind one of the other." I think you mean leaving the odds at 2/3rds that is was behind the other two
nyet if i tossed a coin 300 times and i got heads 300 times
Non` and that all changes when he removes a door
nyet what are the odds i'll get heads again?
Non` the odds completely change
nyet 50/50
Clampy Non, yeah.
nyet it's still a regular coin
Clampy nyet, that's a different kind of problem.
nyet nope
nyet it's not
Fermat nyet - I dunno how you're going to feel when you find out the truth.
dumbier this is why people hate Mensa. not #mensa, Mensa.
Clampy It is.
nyet i'm amazed you don't get this
Clampy The doors he reveals depends on what you picked at first.
nyet only in as much as that it has to be a wrong door
Non` hey Fermat, is there a site that illustrates the answer?
nyet so if you were right or wrong, he still ain't openin your door
Fermat I have no idea
nyet you will ALWAYS come down to making a 50/50 choice between the right door and the wrong door
* gloriae changes topic to 'run for your lives, the 3 doors problem is back'
Clampy nyet, yes, so if you didn't pick the right door, he'll single that door out for you.
dumbier there could be something LESS pointless than this scenario, but then I'd try to start a discussion on it
Non` so you just made this up with your obviously fallible mind?
nyet clampy: and if you did? You don't know if you picked the right door
Clampy Then I'll have my small chance of getting it wrong.
nyet so basically you're just picking out of a dwindling group each time. if you start with four, your odds are 1/4 of getting it right. remove one door
Clampy To another observer that just showed up, it'd look like I had a 50/50 chance.
nyet now you have 1/3 odds of getting it right. remove one door
gloriae I first encountered this problem in a statistics newsgroup in 1986, wherein 2 rival camps were screaming apeshit at each other
Clampy But, you're selecting a door he WON'T reveal, as a start.
gloriae post after post insisting shut up everybody, THIS is the explanation that closes the question
Clampy If the doors are removed at complete random, then you're absolutely correct, the odds don't get better.
Non` Clampy the odds have nothing to do with it at all... you can remove the numbers from the problem... it is a basic theoretical probability problem.... when making a random choice all choices are EQUAL.... am i wrong?
dumbier screaming apeshit... good name for a band
Fermat Yes, Clampy, they do.
gloriae "odds" are theoretical quantitative probability
Non` hi gloriae
Clampy Non, that's not right.
gloriae that's an artificial distinction, like the one you were trying to draw yesterday between 'bail' and 'bond'
Clampy The problem is a unique one. It is plenty confusing.
* Swish[OB] has joined #mensa
nyet` jesus fuckin christ
Non` the question that matters is "does making a distinction between the remaining doors matter? and the answer is no
Clampy Ummm... What if you had a trillion doors. Picked one that he WON'T open, then he opens up every other one?
nyet but that's not the question clampy
Clampy My pick would most assuredly be wrong anyway. So, the second choice would most assuredly be correct.
nyet you always know he'll get down to 2 doors and you'll have to choose between them. So you always come down to a 50/50 chance
Non` oh yes. if you make a choice out of 100 doors you will probably be wrong
Fermat If you switch you have 2 chances in three. If you stick, only chance in three
nyet there's no 'switch' fermat. you're making a choice between 2 doors
nyet one is right, one is wrong
gloriae note that the gameshow host's pick of which door to open was not random
Clampy nyet, the odds are you picked the wrong door in the first place.
Fermat 2 chances if you switch. 1 chance of you stick.
nyet no clampy
dumbier except the one you'll be needing a beating from
Non` but if you narrow it down to 10 doors, with the door you had originally chose still in the mix would you change doors?
nyet it doesn't matter what you picked the first time
gloriae Clampy: that's not statistical probability, that's pessimism
nyet because the situation will ALWAYS be the same in the second choice.
Clampy nyet, if you increase the number of doors to a trillion, it is.
nyet it'll always be a choice between the one right door, and the one wrong door
gloriae the idea is to reduce the situation to quantitative analysis, leaving emotions and extraneous ideas out of it
nyet the *last* choice you're given is 50/50
dumbier I hope this is fermat's last theorem
Fermat I have plenty of others. But they're tougher.
* Clampy suggests making a chart of all possible scenarios.
nyet marilyn vos savant is a damn idiot. clampy
gloriae good evening (morning), macheath
nyet pay attention
Non` right nyet... any unopened door is as good as the next
Clampy Out of a 3 door situation, running the entire scenario.
gloriae 7:44 p.m. in SF
Non` they all could have the car behind them
Fermat Hey gloriae. I love you guys.
Clampy nyet, the history of the little game makes the odds different.
Non` how so? All it does is prove that you weren't wrong
nyet no it doesn't
gloriae the question is: do the odds of picking the right door "change" after one of the doors has been opened and has it become more or less likely that the contestant's first pick was correct
nyet because the game, clampy, IS RIGGED
Non` assuming the host only opens doors that don't have the car behidn it and never opens your door
Non` door*
gloriae Non`: it's an old problem and very carefully stated.
gloriae there's no need for extraneous assumptions
Non` gloriae, did you see Fermat ask it?
Non` "[19:19] Fermat Suppose you're on a gameshow. The prize is a car, behind one of three doors. You choose one, The host opens one of the other doors, and shows you there's no car behind it. Then he says you can choose the other door, if you like. Does it make any difference if you do? If you can work out the right answer in five minutes, then you have a Mensa level I.Q."
ThrasheR rofl
gloriae Fermat Suppose you're on a gameshow. The prize is a car, behind one of three doors. You choose one, The host opens one of the other doors, and shows you there's no car behind it. Then he says you can choose the other door, if you like. Does it make any difference if you do?
gloriae nothing to do with Mensa
ThrasheR exacly
Non` well yeah. but the question is still valid
gloriae most of the people who've wrestled with the problem were and are not in Mensa (most people who qualify aren't in Mensa)
Fermat There's no wrestling. It's obvious to anyone with an I.Q. of greater than about 150.
gloriae and yes, Non`, aside his little Mensa postscript, it's stated correctly (if loosely). It's a probability problem. One needs to know (1) the odds before one door was opened and (2) the odds after one door was opened.
Non` Does it make any difference if choose the other door?
gloriae some people get get emotional about it :)
gloriae but it's like flipping a coin; some people will get emotional about that, and insist that the odds can't be the same every single time, but they are.
Fermat The statement of the problem could be made longer, but it would make no difference.
nyet fermat = assclown. that's pretty obvious
Clampy Heh.
Non` gloriae, i am completely unemotional about this. I just want to know the right answer
Fermat 2/3 chance if you switch, 1/3 if you stick
gloriae I angered a math professor once by refusing to follow his invitations go down that primrose path of believing that previous tosses influenced the outcome of subsequent tosses
gloriae I guess he just wanted it to go on longer, or he'd made a bet that I wouldn't understand it ;P
Clampy glorie, wha..?
gloriae technically, in probability, one either adds or multiplies the 'chances' depending on the type of problem
Fermat I should have thought that would please him, gloriae. The chances of it being a two headed penny, however, do increase. So it isn't that obvious
Non` each time you flip it's a 1/2 chance. the coin has no memory.
gloriae the original probability was 1 in 3 (1/3). One door was opened. The remaining probabity was 1 in 2 (1/2)
ThrasheR exacly
gloriae the trick is to combine those meaningfully. ty ThrasheR :)
gloriae Fermat: that's a red herring, be ashamed of yourself
Fermat wrong, gloriae
ThrasheR i missed pooky
gloriae Fermat: in an honest problem one uses an honest coin and doesn't muddy the waters with presumptions of two headed coins. go to the foot of the class.
Non` gloriae, if you were on the game show and you were asked to pick out of 3 doors would you choose door a, b, or c? humor me
gloriae I must agree with nyet. No thanks, Non`, I'm not humouring anyone this evening.
Fermat I was married today, but Mrs Oracle is a little the worse for Bollinger.
Non` ok nyet are you here? would you choose a,b,or c?
ThrasheR WHAT
Clampy Ummm..
ThrasheR ARE YOU ORACLE?
gloriae Non`: are you looking for 'lucky letters' like some people have lucky numbers?
gloriae of course he is, ThrasheR
Clampy The odds are better to 'switch'.
Fermat Yes. Who's like to see a picture of my blushing bride?
Non` no gloriae, i am going to ask the person if they would change choices after I remove a door
Non` more a psychological experiment
gloriae only if it's on a website, Fermat. for me, anyway
Non` choices*
Fermat I'll send it to anybody who trusts me. It's a jpg file. so it can't do any harm. When was I ever malicious? I'm just a tease.
Clampy One of the ways to solve statistics problems is to draw out every possible scenario, and then counting successes and failures.
Fermat Clampy, it's a losing battle. I've fought it with these guys many a time.
Miriel That's called "brute force"
Non` three hours until Harry Potter! W00T!
Clampy Non, woo.
Fermat You can either form a mental model of the scenario - or you can't.
Clampy Miriel, in a scenario with only 9 possibilities, it's pretty small as 'brute force' goes.
nyet problem with how you're all solving it is that you're pretending you know if you have a success or a failure after each guess. you're forgetting that it's a game show
* minerva has joined #mensa
Clampy Hi minerva.
minerva 3 doors?
nyet it's rigged so that the only choice where you have a chance for success or failure is the last ine.
minerva heyas
* mko94 has joined #mensa
Clampy nyet, the problem, as I know of it, is a tad different.
gloriae yes, that's the brute force method, so-called
Non` i hate to beat a dead horse almost as much as having no closure.
gloriae with a problem this simple, the force is very small and not very brutal :)
Fermat nyet - it makes no difference. The problem is as stated. The game show is irrelevant
gloriae but that's how that approach in math is known
Miriel Oracle: shouldn't you be away ravishing your bride?
nyet no it's not fermat.
Fermat I'll tell you if you let me
gloriae like the kludge in coding
nyet but then again i wouldn't expect you to understand that.
Fermat She's already ravished, and a little the worse for Bollinger, Miriel. I'm finishing a nice bottle of Chardonnay.
nyet wait, Fermat just got married and is on IRC?
nyet i think we can all see who's got the low iq
* arryana sets mode: +b *!*@48.192.pth-ag1.dial.plus.net.uk
vallor nyet: sounds like bs to me
* You were kicked from #mensa by PythonS (Banned)

Private with Nyet
Session Start: Sat Jun 21 04:03:45 2003
Session Ident: nyet
Fermat will you let me tell you the answer?
nyet i'm not interested in your wrong answers, troll.
Fermat I just explained it to Non. Non will tell you. Or I could send you the script.
nyet leave me alone, oracle.
Fermat The fact that you don't want to know the truth explains a lot about your life.
nyet fuck off oracle.
Fermat I think you know.
nyet fuck off, oracle.
Fermat Intellectuals are conceited but they're not pretentious.
nyet you are an asshole and now you are on ignore.

In my experience few people can form a mental model of this problem, even when it's been explained to them. A recent book of puzzles contained the wrong solution. Even mathematicians, who should have known better, have been fooled.

Why such a simple scenario should present mental difficulties is hard to imagine. It proves what a useless tool is common sense.

Okay, here we go.

Does it matter whether the host knows where the car is? Yes, of course it does.

If the host doesn't know where the car is, then one third of the games resolve when he opens the door and finds the car. The other two thirds resolve equally, and it doesn't improve the player's chances to change his mind.

If the host does know there the car is, then in two thirds of the games it's behind one of the two doors that the player didn't choose. The host always opens the door that has no car behind it. Therefore in two thirds of the games the car will be behind the door that the host didn't open.

So, it always doubles the player's chances to change his mind after the host has elininated a door.

Come on, stupid. How hard was that?