Brent Nyitray
Fairlington Blade
Let the mutherfuker burn!
jnc4p
Kevin S. Willis
Michigoose
lmsinca
markinaustin
Arcanaut
novahockey
quarterback
ScottC
yellojktImagine you find yourself transported back in time to the 1970’s and an appearance on Monty Hall’s Let’s Make a Deal, and Monty has just selected you as a contestant. He shows you three doors and assures you that the day’s grand prize is sitting behind one of those doors. If you pick the correct door, the prize is yours. After you pick door number 1, Monty then opens door number 3 to reveal nothing behind it. He then offers you another choice…stick with your original door or switch choices to door number 2. What should you do?
(from Mike)
I hope Scott doesn’t mind my adding to his post. Here’s a link where you can play the game yourself, with both Scott’s and Mark’s scenarios (whether Monty knows where the prize is or not).
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All Things in Moderation 
Ooh, I know. Tell Monty that you'll split the prize with him if he tells you which door you should choose!If I were a mathematician, I'd probably switch doors.If I were a conservative, I'd probably stick to my guns.If I were a liberal, I'd be frozen up debating whether I should change or not.Who else can I make categorical statements about? :-)Thanks for reminding me about the "Monty Hall problem." Haven't thought about it in a few years.
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Well said, Mike. This is a problem that can be used to present the notion of "paradox" to 8th graders.I have never been convinced by the "switch" argument, btw, although I understand it.
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Because Scott will make the "switch" argument, I will make the other.Imagine that the doors are to stalls and there are horses in the stalls. You are told two horses are dead. You pick Door One. Monty opens Door Three: dead meat.There was a 2/3 chance Hall would pick a dead horse and remove it from the mix. Well, he did. So the game has now changed, and there is a one half chance the dead horse is behind each of the remaining doors.
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Mark:I have never been convinced by the "switch" argument, btw, although I understand it. I spent 45 minutes this weekend trying to convince 4 very smart people that switching is the right thing to do. They finally got it in the end. It is not intuitive at all.Imagine that, instead of 3 doors, there were 1,000 doors and you got to pick one. Then Monty opens up 998 empty doors and asks you if you want to switch. Isn't obvious now that you should switch?
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Mark:There was a 2/3 chance Hall would pick a dead horse and remove it from the mix.This is where your argument is wrong. There is not a 2/3 chance he picks a dead horse. There is a 100% chance he picks a dead horse. There is a 2/3 chance that the live horse is behind one of the two unpicked doors, but a 100% chance that at least one of the unpicked doors has a dead horse. And Monty will always show you that one. So there is always a 2/3 chance that the remaining door has the live one.
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Scott, I understand the argument. It is a two person game theory argument based on the assumption Hall KNOWS where the dead horse is.Assume, instead, that he does NOT know and that the two player game is an evenly matched guessing game. Can you see it my way?
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In other words, if Hall is picking blind, you have one result, but if Hall is picking with foreknowledge, you have a different result.
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Mark:Assume, instead, that he does NOT know and that the two player game is an evenly matched guessing game. Can you see it my way?Actually, it doesn't matter whether he knows or not. If he doesn't know, then there is a 1/3 chance that he shows you the door with the prize (rather than a 0% chance), in which case the game is over and you never get a second chance to pick. But if he picks the empty one, even by chance, the remaining one still has a 2/3 probability of being correct. Monty's reason for picking the empty one doesn't change the probability of what is left.
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Think of it this way…Assume your choice is door A. If Monty has foreknowledge, only 3 things can happen:1) The prize is in A and you switch. You lose. 2) The prize is in door B, and you switch. You win. 3) The prize is in door C and you switch. You win. So in 2 out of 3 cases, you win.Now assume Monty has no foreknowledge. 1) The prize is in A. No matter what Monty picks, the game continues. If you switch, you lose. 2) The prize is in B. If Monty chooses B, the game is over, but if he choses C, the game continues. If you switch, you win. 3) The prize is in C. If Monty chooses C, the game is over, but if he chooses B, the game continues. If you switch, you win.So, again, there are more possible outcomes if Monty has no foreknowledge, ie the game can end prematurely in two cases. But in the 3 cases in which the game continues after Monty chooses, you win in 2 cases if you swicth, but only one if you don't. You should always switch.
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And that's why I'm never going to be a contestant on Let's Make a Deal. 😉
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I would just naturally assume Monty knows which door the prize is behind, it's his show after all, and switch. Here's the thing though, I would have switched anyway, but I don't really know what that means. It was just my first gut reaction without thinking through the problem.I've never really thought these kind of games tell you that much about people. I like the ones that are puzzles instead. A great one for kids is:I have two coins that add up to 55 cents and one of them isn't a nickel. What two coins do I have? It's easy but most kids don't get it the first time, especially when it's spoken and not written.
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But one of them IS a nickel.
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Without Hall's foreknowledge, once there are only two doors with one prize the odds are even. The previous choice has no effect on the current choice. Scott, without foreknowledge, this is flipping coins. Flip 19 straight heads and the odds of tails on the next flip are 50-50. Doesn't matter what the odds were against flipping 19 straight heads.Beginning odds here were 2-1. New game, new odds, just like every single coin flip.
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Yep Mark, you're the winner. It seems so obvious to us, but try it on a 10 year old. My husband has tons of those for the kids to get them thinking. But he also reads fortunes off of dominoes so the kids listen to everything he says, even if it doesn't make any sense the first time.
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OTOHIf it is a two person competitive game and Hall knows where the prize is, he only has an incentive to lure you to switch if he knows you have chosen the prize!How about that twist?
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Isn't the Three Door Monty conundrum a Richard Feynman creation? Or am I misremembering?
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Mark:Scott, without foreknowledge, this is flipping coins.No, it isn't. When you choose, I think we agree you have a 1/3 chance of being correct. That means that there is a 2/3 chance that the prize is behind a door you did not pick. But we know, with absolute certainty, that one of those two remaining doors is empty. So there is a 2/3 chance that one has the prize, and a 0% chance that the other has the prize. We just don't know which probability applies to which door. As soon as one is opened and is empty, whether chosen intentionally or not, we now know which door has a 0% probability. Which means the other one must, necessarily, have a 2/3 propability.As I said above, the absence of foreknowledge simply increases the chances that the game ends prematurely, from 0% to 33.33%. But it does not change the probabilities in the cases in which the game does not end prematurely.
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Mark:How about that twist? I like that one. But, in that case, Monty will always, rather than never, open the door with the prize behind it, if you haven't already picked it. That is to say, the only time you will be offered a choice is if you have picked the correct door. So if you assume a Monty with both foreknowledge and an incentive to make you lose, you should never switch.
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Scott- You have me convinced. To reference the common sense post from yesterday, this seems like one of those situations where you have to put aside your "common sense" thoughts.
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Mark:Wikipedia agrees with you…ie if Monty is ignorant, switching does not improve your chances. I think this is wrong, but will ponder it.
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Scott, that 2/3 probability for the remaining doors is the sum of two 1/3 probabilities. Thus when one loser is revealed there is a remaining 1/3 of the original probability behind the door not chosen and the door chosen by the contestant. Now it is 50-50. New two door game.
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Mark:You are right. If Monty is ignorant, that can only reduce your chances of winning by switching, because you are prevented from switching only in cases in which you have chosen wrongly to begin with. If you have chosen correctly, you will always be offered the chance to switch, as before. But if you have chosen incorrectly, half of the time you will not even get the chance to switch when you would have won if you were able to.So it does indeed matter whether Monty is ignorant or not.
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Mike…just saw your addition to the post. That is useful.
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You all lost me with all this switching talk. There's only one thing to know about gambling. bet the pass line and back up your bet to the maximum you can.
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Or find a game that gives the player slightly better odds against the house, such as Spanish 21.
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Is that like double-exposure blackjack? no hole card for the deal. I can't play blackjack b/c I get frustrated with the other players.
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NoVA, you're gone already but no. In Spanish, there are no 10's, face cards yes. The real advantages are being able to draw on 12 or 13 with better odds, automatic pay on any 21 (no matter the dealers cards), ability to draw on split aces, and the biggie is being able to double down on any number of cards, up to 4 times. As always, betting is the key and the player has more opportunity to have more money on the table. Of course, it also increases the chances of losing a lot of money. It's not a game for the faint-hearted, unless you play it just like regular black jack, which defeats the purpose of playing spanish. I like watching the other players chicken out, but most players who play it are pretty experienced black jack players so there's less inconsistency and more adrenalin.
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Here's a funny story about Spanish 21. The players don't generally want someone to sit down and play who's never played before. You'd be surprised how many people watch for a minute and if there's an open chair will ask "Do you have to speak Spanish to play?", it's a bit of an underground tradition to say "si" and then those of us who speak a little Spanish carry on with that theme and the person usually wanders off. Of course everyone cracks up once they're out of earshot. Even the dealers play along sometimes, usually just by keeping quiet though. It's mean I know, but funny still.
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You always switch doors. Your original decision was a 1 in 3 call. Monty is asking you if you were right the first time or wrong. The odds are 2 in 3 that you were wrong.Here's one way of gaming it out. You play the game with Monty 1000 times and always pick Door A. Monty always reveals what was behind B or C. If the big prize was behind A, you lose. If the big prize is behind B or C, you win. 2 in 3. Switch.Incidentally, if Monty doesn't know where the prize is located, then half of the time, he'll open a door with a prize behind it. Of course, you should switch in that case. BB
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"You always switch doors. Your original decision was a 1 in 3 call. Monty is asking you if you were right the first time or wrong. The odds are 2 in 3 that you were wrong."Math makes my brain hurt.
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Here's a fun variant. Let's go to MontyLand. 1000 doors! 999 with a raw egg and one with a golden egg. You pick one door and then Monty opens 998 ones with a raw egg.If you think that you got it right that first time, then stick with it by all means. If you think you didn't get the 1 in 1000 long shot, then switch. Judging from the number of times I've watched Deal or No Deal, a lot of people would take that bet.BB
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