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Post by omaha on Jun 17, 2009 12:11:56 GMT -5
I think you'd need to ask Carol Merrill about that.
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Post by millring on Jun 17, 2009 12:19:39 GMT -5
You are almost there. You are exactly right that Monty is being theatrical. But you are slicing things wrong. It boils down to this: You are offered a choice of one door out of three (1/3 odds). You are then offered the choice of taking two doors out of three (2/3). Ignore all the theatrics taking place between the two choices. You can either pick one door and take your chances, or you can pick two doors and take your chances. My point was nothing about theatrics. You were offered one of the doors with 1/3 odds. Each door had the same odds before the reveal -- though the game was set up such that you will eventually see that in reality it was ALWAYS going to be a 50/50 game. Nothing about the third door (not your first choice, nor Monte's reveal) EVER had any more than a 50/50 chance of being right. Bundling it with Monte's goat door did not suddenly confer upon it a 2/3 odds. It NEVER had 2/3 odds. It was initially presented as one of three 1/3 doors. After the reveal you then realize that it and your door were ALWAYS merely two doors of a 50/50 proposition. Further, NOTHING about Monte's revealed door made your initial choice of door wrong, nor changed it from the 50/50 choice with which you may not have understood going in, but after the reveal realized. |
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Post by omaha on Jun 17, 2009 12:23:10 GMT -5
Ok, lets make it simpler.
There are three doors. One contains a prize.
Monty says "pick any two. If the prize is behind one of the two, you win".
That's what this game comes down to.
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Post by millring on Jun 17, 2009 12:35:15 GMT -5
Ok, lets make it simpler. There are three doors. One contains a prize. Monty says "pick any two. If the prize is behind one of the two, you win". That's what this game comes down to. Perhaps, but that's not what's "counterintuitive" about the dilemma as explained. The sticking point is in the assertion that after Monte revealed wrong choice, changing your choice from your initial choice to the one Monte didn't reveal increases your odds of winning. |
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Post by aquaduct on Jun 17, 2009 12:45:09 GMT -5
Millring,
If it's any comfort, this is what got the Vos Savant woman death threats from mathemeticians.
If you assume the game starts over after the first door is revealed, it's a simple 50/50 shot and your first pick means squat.
But, as a gambler, the game doesn't start over. Your first choice is meaningful. You pick your first door entirely at random. But Monte's action introduces outside information into the problem eliminating randomness. That changes your strategy.
It's similar to playing blackjack. In isolation, each hand has certain odds. But those odds change if you're counting cards. Which, if you care more about winning money than math, forces you to change strategies.
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Post by millring on Jun 17, 2009 13:20:07 GMT -5
But, as a gambler, the game doesn't start over. Your first choice is meaningful. You pick your first door entirely at random. But Monte's action introduces outside information into the problem eliminating randomness. I'm really not trying to be contrary, but I really just don't get what Monte's action does that "introduces outside information" to the other two picks (my original pick and the one he didn't reveal). They were always going to be my two 50/50 choices. Just because Monte revealed one of the goats, that in no way "introduces outside information" about either my initial choice or the other door, except that one of them is a winner and one is a loser. So there's no advantage to be had in changing my choice from my initial one to the one Monte didn't reveal. Blackjack is different. If you can count the cards that have passed by, you know they will not come up again and you thus know how your odds are changing -- either for or against your current cards. |
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Post by Fingerplucked on Jun 17, 2009 13:52:25 GMT -5
I'm really not trying to be contrary, but I really just don't get what Monte's action does that "introduces outside information" to the other two picks (my original pick and the one he didn't reveal). They were always going to be my two 50/50 choices. Ok, let's try it this way. There are three doors. Behind two of them are goats. Behind one of them is the car. You pick one. The Monty opens a door. And since he knows where the car is, the door he opens will always contain a goat. BUT, instead of doors, let's use windows. Big, wide, windows. I think it will all be clear if you can see what's inside each little room as the process unfolds. Ready? Monty tells you that there are three rooms, one each behind the three windows. You look and you see the rooms. Monty tells you that one and only one of the rooms contains a car, and that the other two contain goats. You look, and behind the windows you see two goats and a shiny new car. Monty tells you to pick a window, any window. You choose window #1. Now Monty tells you that he doesn't know if the car is behind window #1 or not. WTF? you say to yourself. You can plainly see that the car is behind window #2. You didn't know that you were supposed to pick the room with the car. Monty said "pick a window, any window," and you did. He tricked you. The bastard. You're mad at him and you're mad at yourself. But now Monty begins to open the window to room #3. Once the window is fully open, Monty says "It's a goat!" Dumb shit, you're thinking. I could already see that it's a goat.To your surprise, Monty now asks you if you'd like to change your guess from window #1 to window #2. You look at the car behind window #2, then you ask Monty to repeat the question, wanting to make sure that you understand . . . . See how clear it is that your odds are better than 50/50 if you always change your guess, provided that Monty's using windows? |
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Post by millring on Jun 17, 2009 13:55:46 GMT -5
To quote Paul Schlimm:
::nods::
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Post by millring on Jun 17, 2009 13:59:01 GMT -5
I never metaphysical
I couldn't be
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Post by omaha on Jun 17, 2009 14:27:23 GMT -5
In the end, all this comes down to is Monty saying "Pick two doors. If the prize is behind either of them, you win."
The fact that he leads you through all the theatrics is just smoke and mirrors and misdirection. You pick two doors out of three. If the prize is behind one of your two doors, you win.
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Post by millring on Jun 17, 2009 14:37:06 GMT -5
In the end, all this comes down to is Monty saying "Pick two doors. If the prize is behind either of them, you win." The fact that he leads you through all the theatrics is just smoke and mirrors and misdirection. You pick two doors out of three. If the prize is behind one of your two doors, you win. But what does that have to do with the notion put forth that your odds get better if you change your pick after Monte reveals one goat? |
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Post by omaha on Jun 17, 2009 14:39:52 GMT -5
There are only two ways to play the game: Switch or don't switch.
If you don't switch, your odds of winning are 1/3.
If you do switch, your odds of winning are 2/3.
So, its in that sense that your odds "get better" by switching, but that's not how I would put it. I would put it this way: By switching, you are abandoning a strategy with a 1/3 chance of winning, in favor of a strategy with a 2/3 chance of winning.
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Post by millring on Jun 17, 2009 14:41:41 GMT -5
If you do switch, your odds of winning are 2/3. In what way did the odds become 2/3 by switching choices? |
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Post by omaha on Jun 17, 2009 14:42:53 GMT -5
Ok, you're just screwing with me, right?
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Post by Fingerplucked on Jun 17, 2009 14:43:45 GMT -5
I'm tellin' you, windows. They're the eyes to the soul.
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Post by millring on Jun 17, 2009 14:45:32 GMT -5
Ok, you're just screwing with me, right? No, I'm not. Nothing has logically been established except that the choice, the game, was always going to be a 50/50 choice. And when you have a 50/50 choice there's never any statistical advantage to either switching OR keeping the same choice. |
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Post by TDR on Jun 17, 2009 14:53:35 GMT -5
Well, Millring, rest assured no one will ever convince me of your density. The fact is, this is damn hard to explain properly. I'll try this.... Forget everything that happens AFTER you make the initial choice. At that time ALL the doors are divided into two universes: 1. The door you picked (with it's 1/3 chance of being correct); and 2. The doors you didn't pick, with their 2/3 chance of containing the prize. Now, as you know, Monte did nothing when he opened the other door, because he KNEW it was a goat door. NOTHING CHANGED. Just a clever trick, smoke and mirrors, basically, that does not change the odds of either universe. Your universe STILL has it's 1/3 chance, and the remaining door still has it's 2/3 chance. When he offers you the chance to change your pick, he's essentially offering to let you choose the entire universe of doors you did not pick initially (by eliminating the goat door, that is now out of play, so the entire 2/3 chance of that universe is now held by the remaining door you did not pick, initially). If any of you know a really competant gambler, try this on them. I'm not talking about the casual poker player, but the type of guy who can make a living at blackjack or dice. They may or may not be educated, but it has amazed me that they tend to GET the right answer on this, while folks who have actually have math degrees do not. For whatever reason, it's NOT counterintuitive to a gambler. Anyway, the way a gambler explained his reasoning to me was: Monte is essentially giving you the chance to pick ALL the doors you originally excluded. In the 100-door scenario, monte has opened 98 of the doors you did NOT pick, leaving one closed among the whole universe of doors you did NOT choose. THAT single closed door you did not choose now has enormous odds of being THE door. And the single door you initially chose still has it's lousy 1% chance of being THE door. Gotta change my pick, Monte. I'm with Millring on this one. Vos Savant and all the other geniuses notwithstanding. I don't see Fingerplucked or Omaha have made the convincing case yet. After Monte reveals a door that has no car, its a new game with a 50/50 chance of being right. Bet heads or tails on a coin toss, it makes zero difference what came up on the previous toss or the previous 100 tosses. Odds are still 50/50. Yes Monte showed you a goat. Odds are still 50/50 now on which door has the car behind it. How is your information any better given what has gone before? Monte is essentially giving you the chance to pick ALL the doors you originally excluded. In the 100-door scenario, monte has opened 98 of the doors you did NOT pick, leaving one closed among the whole universe of doors you did NOT choose. THAT single closed door you did not choose now has enormous odds of being THE door.I don't know, doesn't this presuppose Monte is intentionally leading you to change your guess by offering up the one door that he didn't choose and you didn't choose? But is he? Where's the indication that you chose the right door, if you did? If you follow this logic and change your guess, you're just as likely to change from right to wrong as to get it right on your second try. |
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Post by Supertramp78 on Jun 17, 2009 14:57:22 GMT -5
let me put it this way. Without Monty opening any door, the odds of you winning are always one in three. We can all agree to that. So lets set up three doors. For the purposes of this explanation we will call these doors CAR, DONKEYA, DONKEYB
There are three possible initial setups. You have picked either car OR donkeyA OR donkeyB. With me so far?
IF you pick the car, Monty comes along and opens either DonkeyA or Donkeyb and offers you the choice and you take it. this moves you from a car to a donkey. This sucks. So if you start on door CAR, you LOSE.
IF you pick donkeyA, Monty comes along and has to show you DonkeyB (the only other non winning door on the floor). You switch and get CAR. You WIN.
IF you pick donkeyB, Monty comes along and has to show you DonkeyA (the only other non winning door on the floor). You switch and get CAR. You WIN.
So from any starting point where your odds were intially one in three, the odds of winning after a switch become, let's count (lose, win, win), two in three. Those odds are better.
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Post by billhammond on Jun 17, 2009 14:57:49 GMT -5
I can't make these explanations work in my addled brain, either.
No matter how I diagram the sentences, I don't see how the odds change.
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Post by TDR on Jun 17, 2009 15:03:02 GMT -5
There are only two ways to play the game: Switch or don't switch. If you don't switch, your odds of winning are 1/3. If you do switch, your odds of winning are 2/3. So, its in that sense that your odds "get better" by switching, but that's not how I would put it. I would put it this way: By switching, you are abandoning a strategy with a 1/3 chance of winning, in favor of a strategy with a 2/3 chance of winning. I don't think so, Jeff. Look at it this way. Monte says, "The way this usually works is you pick one door in three. But because you're special, I'm gonna toss out one of the goat doors right off and give you a 50/50 shot at this car". Explain how your chances just went from 1/3 to 2/3. Now suppose you had a hunch the car was behind door #1. Doors 1 and 2 are left. One has the car, it might be either. How does what your hunch was affect the odds, or change them from 50/50/ to 2/3? What has gone before is irrelevant. Other than you chances have changed from 1/3 to 1/2. The likelyhood that you guessed correctly the first time is entirely unchanged. |
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