# Deal Or No Deal/Monty Hall Problem **URL:** \ **Category:** Factual Questions\ **Created:** [March 5, 2006, 8:47am UTC](https://boards.straightdope.com/t/deal-or-no-deal-monty-hall-problem/347123 "2006-03-05T08:47:57Z")\ **Posts on this page:** 20\ **Page:** 3
**Author:** ![FeelScknd](https://avatars.discourse-cdn.com/v4/letter/f/3e96dc/32.png) [@FeelScknd](https://boards.straightdope.com/u/FeelScknd)\ **Post date:** [January 7, 2013, 5:52pm UTC](https://boards.straightdope.com/t/deal-or-no-deal-monty-hall-problem/347123/41 "2013-01-07T17:52:49Z")
> [@Great\_Antibob](#): > > You could not be more wrong. > > If Monty doesn’t absolutely know where the goat is, he opens the door with the car some fraction of the time (actually 1/3 of the time). > > But this never happens, because Monty KNOWS where the car is and NEVER opens that door. > > So, yes, the point IS that Monty knows where the car is, and that affects the probability. It affects the probability that he would open the door with the car (as you said, it would happen 1/3 of the time). The times he does, the game is over and the problem doesn’t come into play. The participants chances were always 1/3 The 2/3 of the time he doesn’t would be physically identical to the problem where he intentionally opened a door with a goat behind it. In these 2/3 of the time the problem is actually allowed to occur and becomes a subcategory upon which we base the likelihood of the location of the car are the same since we’re dealing with same results. This is where things get confusing for me and everybody else You’re all right in saying I don’t see how this subcategory could be any different than th formal version of the problem. I don’t grasp how his KNOWING and opening the wrong door intentionally somehow confers a different probability. By breaking down into a subcategory he still opens the wrong door every single time within this category, the only time the full game is played. The options and possible permutations are still the same where one possibility is he switches and gets a goat and the other two possibilities he switches and gets a car–as broken down graphically in the “simple solutions” portion on the Wiki page for the problem. It would still appear to me that including the times where the game is over early (and we could therefore consider the times it’s not as a new and separate game) is to also consider how often a contestant wearing a blue shirt wins as statistically relevant. I don’t see it bearing any influence on the new game any more than… what’s that logical fallacy that relates to flipping a coin? No matter how many times you flip it, though odds are it should turn up say “heads” eventually, each flip is it’s own and the odds, possibility remains only 1/2. Different logic problem, unfortunately brings up the 1/2 stat again which could lead to confusion, but I hope it illustrates better where I’m coming from. ---
**Author:** ![FeelScknd](https://avatars.discourse-cdn.com/v4/letter/f/3e96dc/32.png) [@FeelScknd](https://boards.straightdope.com/u/FeelScknd)\ **Post date:** [January 7, 2013, 6:02pm UTC](https://boards.straightdope.com/t/deal-or-no-deal-monty-hall-problem/347123/42 "2013-01-07T18:02:16Z")
> [@Thudlow\_Boink](#): > > Since you’re new here, **FeelScknd** , you should know that the Monty Hall problem has been discussed in great detail here in previous threads, such as [this one](http://boards.straightdope.com/sdmb/showthread.php?t=329774) and [this one](http://boards.straightdope.com/sdmb/showthread.php?t=527671), which contains links to yet others and to the Straight Dope column(s) in which Cecil Adams gave us the Straight Dope on the problem. The chances of your bringing us any new enlightenment on the problem are small. Thank you for the heads up. I appreciate it. Wanted to get my head working this morning. I’d forgotten all about MHP or straight dope but stumbled into it. I assumed and from what I’ve read it’s been done to death. Didn’t realize I’d bump a whole thread out of the dark ages for all to se by working he problem out and posting–forgotten how forums work… comment on blogs more these days. You’re right to say something. I doubt this is where I’ll make my shining contribution to academics or outdo vos Savant on the matter. Must get tiresome. Withdrawn. ---
**Author:** ![Ludovic](https://avatars.discourse-cdn.com/v4/letter/l/7ab992/32.png) [@Ludovic](https://boards.straightdope.com/u/Ludovic)\ **Post date:** [January 7, 2013, 6:04pm UTC](https://boards.straightdope.com/t/deal-or-no-deal-monty-hall-problem/347123/43 "2013-01-07T18:04:10Z")
> [@Superhal](#): > > I thought Monty Hall posts were automatically locked and thrown into the deepest dungeon possible? No, that’s Monty **Haul** posts that are dungeon-related. ---
**Author:** ![TATG](https://avatars.discourse-cdn.com/v4/letter/t/50afbb/32.png) [@TATG](https://boards.straightdope.com/u/TATG)\ **Post date:** [January 7, 2013, 6:23pm UTC](https://boards.straightdope.com/t/deal-or-no-deal-monty-hall-problem/347123/44 "2013-01-07T18:23:30Z")
> [@FeelScknd](#): > > This is where things get confusing for me and everybody else > > You’re all right in saying I don’t see how this subcategory could be any different than th formal version of the problem. I don’t grasp how his KNOWING and opening the wrong door intentionally somehow confers a different probability. By breaking down into a subcategory he still opens the wrong door every single time within this category, the only time the full game is played. The options and possible permutations are still the same where one possibility is he switches and gets a goat and the other two possibilities he switches and gets a car–as broken down graphically in the “simple solutions” portion on the Wiki page for the problem. I’m assuming the way this works is that the game ends if Monty reveals the prize. 3 doors, A,B,C. Assume A is the winning door. Pick A: Monty Picks B, switch loses Pick A: Monty Picks C, switch loses Pick B: Monty Picks A, game over Pick B: Monty Picks C, switch wins Pick C: Monty Picks A, game over Pick C: Monty Picks B, switch wins Half the time when we pick wrong, the game ends early. But every time we pick correctly, the game does not end early. In this case, if we are given the option to switch, there is a 1 in 2 chance that we started off by choosing the winning option to begin with. Consider a variant where we have a goat, an ass, and a car. If we are still in the game, the probability that we picked the goat, ass, and car are not the same. ---
**Author:** ![Great\_Antibob](https://avatars.discourse-cdn.com/v4/letter/g/e47c2d/32.png) [@Great\_Antibob](https://boards.straightdope.com/u/Great_Antibob)\ **Post date:** [January 7, 2013, 6:34pm UTC](https://boards.straightdope.com/t/deal-or-no-deal-monty-hall-problem/347123/45 "2013-01-07T18:34:44Z")
> [@FeelScknd](#): > > It affects the probability that he would open the door with the car (as you said, it would happen 1/3 of the time). > > The times he does, the game is over and the problem doesn’t come into play. > The participants chances were always 1/3 That changes the terms of the game (btw, I’m not sure what point **CurtC** is trying to make on the difference between the actual TV show and the probability exercise). As the game is traditionally presented, Monty NEVER opens the door with the car. What purpose is served by changing the game so it is entirely different? As originally presented, Monty knowing what’s behind the doors isn’t the important thing. The important thing is that he NEVER opens a door with the car. Whether that’s accomplished because he has foreknowledge or because he gets signaled by the producers or through magic, it doesn’t matter. It’s NOT the same game if he does sometimes open a door with the car and the game ends there, because that doesn’t ever happen in the traditional “Monty Hall Problem”. Monty is guaranteed, under the terms of the game, never to do that. Again, it doesn’t matter “how” this is accomplished, only that it does. Basically, you want a variation on the game where the key factor (Monty never opening the door with the car) is no longer part of the game but also doesn’t count towards the probability. That’s a radical change that is fundamentally different from the original version. ---
**Author:** ![TATG](https://avatars.discourse-cdn.com/v4/letter/t/50afbb/32.png) [@TATG](https://boards.straightdope.com/u/TATG)\ **Post date:** [January 7, 2013, 6:38pm UTC](https://boards.straightdope.com/t/deal-or-no-deal-monty-hall-problem/347123/46 "2013-01-07T18:38:57Z")
(In case it wasn’t clear, my post was an attempt to explain what the implications are just if Monty can open the winning door. I wasn’t assuming that in the normal problem he can so do.) ---
**Author:** ![Telemark](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/telemark/32/372_2.png) [@Telemark](https://boards.straightdope.com/u/Telemark)\ **Post date:** [January 7, 2013, 6:56pm UTC](https://boards.straightdope.com/t/deal-or-no-deal-monty-hall-problem/347123/47 "2013-01-07T18:56:03Z")
> [@FeelScknd](#): > > It affects the probability that he would open the door with the car (as you said, it would happen 1/3 of the time). No, this scenario NEVER happens. That is the whole point of the set up, MH knows where the car is and never opens that door. If you ignore that point then you’ve set up a different problem that may be interesting, but isn’t the Monty Hall problem. ---
**Author:** ![CurtC](https://avatars.discourse-cdn.com/v4/letter/c/ce73a5/32.png) [@CurtC](https://boards.straightdope.com/u/CurtC)\ **Post date:** [January 7, 2013, 7:11pm UTC](https://boards.straightdope.com/t/deal-or-no-deal-monty-hall-problem/347123/48 "2013-01-07T19:11:46Z")
> [@FeelScknd](#): > > You’re all right in saying I don’t see how this subcategory could be any different than th formal version of the problem. I don’t grasp how his KNOWING and opening the wrong door intentionally somehow confers a different probability. Here’s an example that should make this difference crystal clear. Imagine that you’re playing against “Evil Monty,” who wants you to lose, and only offers you the choice of whether to switch if you’ve picked correctly with your first guess. In this case, the fact that he has even offered the choice tells you that you had better stick with your first guess. ---
**Author:** ![TriPolar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tripolar/32/3008_2.png) [@TriPolar](https://boards.straightdope.com/u/TriPolar)\ **Post date:** [January 7, 2013, 7:35pm UTC](https://boards.straightdope.com/t/deal-or-no-deal-monty-hall-problem/347123/49 "2013-01-07T19:35:39Z")
> [@FeelScknd](#): > > It affects the probability that he would open the door with the car (as you said, it would happen 1/3 of the time). > > The times he does, the game is over and the problem doesn’t come into play. > The participants chances were always 1/3 Monty never picks the door with the prize. The game goes like this: Three doors. Contestant selects a door. Monty reveals one of the other two doors,\*\* that does not have a prize behind it.\*\* Contestant can either have what is behind the door they selected, or what is behind the third door which was not selected or revealed. Two out of three times the prize is behind one of the doors the contestant did not select initially. In every game Monty reveals one of those two doors that does not have a prize. That means two out of three times the third door has the prize. That’s why it’s better to switch. Another way to look at it is this. At least one of the two doors not selected initially has no prize behind it. Therefore switching is like getting both of the prizes that are behind both of the doors not initially selected. One out of three times neither of those doors has a prize behind it, but two out of three times the prize is behind one of those doors. ---
**Author:** ![CurtC](https://avatars.discourse-cdn.com/v4/letter/c/ce73a5/32.png) [@CurtC](https://boards.straightdope.com/u/CurtC)\ **Post date:** [January 7, 2013, 8:55pm UTC](https://boards.straightdope.com/t/deal-or-no-deal-monty-hall-problem/347123/50 "2013-01-07T20:55:38Z")
> [@TriPolar](#): > > The game goes like this: > > Three doors. > Contestant selects a door. > Monty reveals one of the other two doors,\*\* that does not have a prize behind it.\*\* > Contestant can either have what is behind the door they selected, or what is behind the third door which was not selected or revealed. There have been enough discussions on this in the past, that we shouldn’t be leaving vague statements like this hanging out there. In your bolded part above, did Monty open a door without a prize because a) he has to, those are the rules that he always has to open a non-winning door, or b) because he picked a door at random and in this particular case that happened to reveal a non-winning prize? To state the puzzle properly, you have to be specific that Monty is constrained that he must open the non-winning door because he knows which door is the winning one and he has to open another one. ---
**Author:** ![TriPolar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tripolar/32/3008_2.png) [@TriPolar](https://boards.straightdope.com/u/TriPolar)\ **Post date:** [January 7, 2013, 9:18pm UTC](https://boards.straightdope.com/t/deal-or-no-deal-monty-hall-problem/347123/51 "2013-01-07T21:18:33Z")
> [@CurtC](#): > > There have been enough discussions on this in the past, that we shouldn’t be leaving vague statements like this hanging out there. > > In your bolded part above, did Monty open a door without a prize because a) he has to, those are the rules that he always has to open a non-winning door, or b) because he picked a door at random and in this particular case that happened to reveal a non-winning prize? > > To state the puzzle properly, you have to be specific that Monty is constrained that he must open the non-winning door because he knows which door is the winning one and he has to open another one. It is a requirement of both the logical problem and the actual TV show. He must open a door in every game, and that door cannot have the prize behind it. On the TV show that means it can’t have the grand prize behind it. In the logical problem there is just one prize and two empty doors. But it is required that one empty door is revealed. ---
**Author:** ![Maserschmidt](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/maserschmidt/32/18829_2.png) [@Maserschmidt](https://boards.straightdope.com/u/Maserschmidt)\ **Post date:** [January 7, 2013, 9:23pm UTC](https://boards.straightdope.com/t/deal-or-no-deal-monty-hall-problem/347123/52 "2013-01-07T21:23:55Z")
We should have a Monty Hall/Conveyor Belt SubForum to which this and similar threads can be moved. At the top, the most representative threads would be stickied. ---
**Author:** ![TriPolar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tripolar/32/3008_2.png) [@TriPolar](https://boards.straightdope.com/u/TriPolar)\ **Post date:** [January 7, 2013, 9:42pm UTC](https://boards.straightdope.com/t/deal-or-no-deal-monty-hall-problem/347123/53 "2013-01-07T21:42:04Z")
> [@Maserschmidt](#): > > We should have a Monty Hall/Conveyor Belt SubForum to which this and similar threads can be moved. At the top, the most representative threads would be stickied. It’ll never fly. ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [January 7, 2013, 9:55pm UTC](https://boards.straightdope.com/t/deal-or-no-deal-monty-hall-problem/347123/54 "2013-01-07T21:55:40Z")
> [@Maserschmidt](#): > > We should have a Monty Hall/Conveyor Belt SubForum to which this and similar threads can be moved. At the top, the most representative threads would be stickied. We could call it “The Shitstorm” ---
**Author:** ![Maserschmidt](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/maserschmidt/32/18829_2.png) [@Maserschmidt](https://boards.straightdope.com/u/Maserschmidt)\ **Post date:** [January 7, 2013, 10:00pm UTC](https://boards.straightdope.com/t/deal-or-no-deal-monty-hall-problem/347123/55 "2013-01-07T22:00:18Z")
> [@TriPolar](#): > > It’ll never fly. Well of course it will, even if you posit massless - hey,waaaaaaait a minute. :dubious: ---
**Author:** ![CurtC](https://avatars.discourse-cdn.com/v4/letter/c/ce73a5/32.png) [@CurtC](https://boards.straightdope.com/u/CurtC)\ **Post date:** [January 7, 2013, 10:07pm UTC](https://boards.straightdope.com/t/deal-or-no-deal-monty-hall-problem/347123/56 "2013-01-07T22:07:36Z")
> [@TriPolar](#): > > It is a requirement of both the logical problem and the actual TV show. The TV show did NOT work that way. Way back when there was [an article in the NY Times](http://www.nytimes.com/1991/07/21/us/behind-monty-hall-s-doors-puzzle-debate-and-answer.html) about the problem, where they actually interviewed Monty Hall and got him to participate in testing what the real answer was. Here is their description of the result: > [@NY Times](#): > > On the first, the contestant picked Door 1. > > “That’s too bad,” Mr. Hall said, opening Door 1. “You’ve won a goat.” > > “But you didn’t open another door yet or give me a chance to switch.” > > “Where does it say I have to let you switch every time? I’m the master of the show. Here, try it again.” > > On the second trial, the contestant again picked Door 1. Mr. Hall opened Door 3, revealing a goat. The contestant was about to switch to Door 2 when Mr. Hall pulled out a roll of bills. > > “You’re sure you want Door No. 2?” he asked. “Before I show you what’s behind that door, I will give you $3,000 in cash not to switch to it.” > > “I’ll switch to it.” > > “Three thousand dollars,” Mr. Hall repeated, shifting into his famous cadence. “Cash. Cash money. It could be a car, but it could be a goat. Four thousand.” > > “I’ll try the door.” > > “Forty-five hundred. Forty-seven. Forty-eight. My last offer: Five thousand dollars.” > > “Let’s open the door.” > > “You just ended up with a goat,” he said, opening the door. > > The Problem With the Problem > > Mr. Hall continued: “Now do you see what happened there? The higher I got, the more you thought the car was behind Door 2. I wanted to con you into switching there, because I knew the car was behind 1. That’s the kind of thing I can do when I’m in control of the game. You may think you have probability going for you when you follow the answer in her column, but there’s the pyschological factor to consider.” > > He proceeded to prove his case by winning the next eight rounds. Whenever the contestant began with the wrong door, Mr. Hall promptly opened it and awarded the goat; whenever the contestant started out with the right door, Mr. Hall allowed him to switch doors and get another goat. The only way to win a car would have been to disregard Ms. vos Savant’s advice and stick with the original door. > > Was Mr. Hall cheating? Not according to the rules of the show, **because he did have the option of not offering the switch, and he usually did not offer it.** > > And although Mr. Hall might have been violating the spirit of Ms. vos Savant’s problem, he was not violating its letter. Dr. Diaconis and Mr. Gardner both noticed the same loophole when they compared Ms. vos Savant’s wording of the problem with the versions they had analyzed in their articles. > > “The problem is not well-formed,” Mr. Gardner said, “unless it makes clear that the host must always open an empty door and offer the switch. Otherwise, if the host is malevolent, he may open another door only when it’s to his advantage to let the player switch, and the probability of being right by switching could be as low as zero.” Mr. Gardner said the ambiguity could be eliminated if the host promised ahead of time to open another door and then offer a switch. ---
**Author:** ![etv78](https://avatars.discourse-cdn.com/v4/letter/e/d78d45/32.png) [@etv78](https://boards.straightdope.com/u/etv78)\ **Post date:** [January 7, 2013, 10:11pm UTC](https://boards.straightdope.com/t/deal-or-no-deal-monty-hall-problem/347123/57 "2013-01-07T22:11:41Z")
Putting aside my annoyance at reanimation, I have a question: Is the fact Howie explicitly says at the top of the show he doesn’t know where the $1M is relavent? Or is that the whole point? ---
**Author:** ![TriPolar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tripolar/32/3008_2.png) [@TriPolar](https://boards.straightdope.com/u/TriPolar)\ **Post date:** [January 7, 2013, 10:47pm UTC](https://boards.straightdope.com/t/deal-or-no-deal-monty-hall-problem/347123/58 "2013-01-07T22:47:28Z")
> [@CurtC](#): > > The TV show did NOT work that way. Way back when there was [an article in the NY Times](http://www.nytimes.com/1991/07/21/us/behind-monty-hall-s-doors-puzzle-debate-and-answer.html) about the problem, where they actually interviewed Monty Hall and got him to participate in testing what the real answer was. Here is their description of the result: Ok, I’ve seen some of the stuff Monty has said, and differently from the logical problem, he does try to coerce contestants to switch doors. That is not a factor in the logical problem, there are no incentives in that, and I wasn’t trying to say the rules were all the same for both the TV show and the question, just that an empty door is always revealed. Now your article does not indicate Monty ever did deny someone an opportunity to switch on an actual show, so I don’t know if the show ever did anything but offer the chance to switch, though I do remember some of the earliest episodes and the format hadn’t been so well established yet, so maybe there were some exceptions. But that is not part of the logical problem. It’s really a pretty simple thing, that just confuses people based on pre-conceived ideas. Some people don’t realize that there’s always an empty door revealed. It’s confusing to talk about odds changing when actually your just switching to a different situation where the odds are different. But if you follow the rules of the logical problem, something you can do with a pencil and paper, or three playing cards, or just write a program if you can, you’ll see that ‘switching’ will get you a win two out of threes times, while sticking with your original selection wins only once out of three times. To me the easiest way to understand this is to consider that by having an empty door revealed you are effectively being given the choice of taking what is behind both of the doors you didn’t select, or just the one door you did select. It should be obvious that you have better odds taking what is behind two doors instead of one. ---
**Author:** ![BigT](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bigt/32/12044_2.png) [@BigT](https://boards.straightdope.com/u/BigT)\ **Post date:** [January 8, 2013, 7:57pm UTC](https://boards.straightdope.com/t/deal-or-no-deal-monty-hall-problem/347123/59 "2013-01-08T19:57:28Z")
> [@etv78](#): > > Putting aside my annoyance at reanimation, I have a question: Is the fact Howie explicitly says at the top of the show he doesn’t know where the $1M is relavent? Or is that the whole point? Not really. That’s relevant in the actual game, because Howie could purposefully or even accidentally influence you to pick or not pick the right one (depending on his preference), but it’s irrelevant in the hypothetical game where the host does not not interact with the player except at the proscribed time (i.e. when asking them to pick a case and asking them to switch). The actual place you might be able to get information from is the Banker, where, if you can do the expected value calculation yourself, you can see if he’s overvaluing it. But even that’s pretty iffy, as the Banker doesn’t need to know what’s in your case to do his job, and he can always be malevolent and offer you more because he knows you’re not going to take it. Plus they have some pre-priced prizes that they introduce, which throws the buyout price off even more. It is somewhat fascinating how they somehow took “Pick a number between 1 and 26,” and made it fairly compelling television. (or if you must include the switch, it would be “Pick two numbers between 1 and 26. Now pick one of those numbers.” ---
**Author:** ![Learjeff](https://avatars.discourse-cdn.com/v4/letter/l/94ad74/32.png) [@Learjeff](https://boards.straightdope.com/u/Learjeff)\ **Post date:** [January 8, 2013, 11:10pm UTC](https://boards.straightdope.com/t/deal-or-no-deal-monty-hall-problem/347123/60 "2013-01-08T23:10:39Z")
> [@TriPolar](#): > > To me the easiest way to understand this is to consider that by having an empty door revealed you are effectively being given the choice of taking what is behind both of the doors you didn’t select, or just the one door you did select. It should be obvious that you have better odds taking what is behind two doors instead of one. Whenever you’re given the opportunity to switch, it’s clear that your original guess had a 33% chance of being right, and the remaining door has a 67% chance of being right. Of course, if Monty always shows you that you chose the goat right away, that logic doesn’t hold, and you’d always stick with your guess. Another thing that’s ignored above is that in the game, there were always 1 goat, one good prize, and one grand prize. If Monty shows the goat, you know you have one or the other. But sometimes he shows the good prize, and gives you the chance to switch. Most people don’t. IIRC, this often happened even when they’d chosen the goat. So, Monty doesn’t always show you the goat if you pick it. In any case, I don’t think it bears on the Deal or No Deal situation, since the player picks the numbers, and not an MC who knows what the truth is. That is further complicated by the number of different prizes, and the role of the banker. I have often been surprised by the banker’s offers (both on the low and high sides), so I suspect the banker adds a random component, to avoid being too predictable. Or maybe my mental math was wrong; it often is. [Previous page](https://boards.straightdope.com/t/deal-or-no-deal-monty-hall-problem/347123.md?page=2) [Next page](https://boards.straightdope.com/t/deal-or-no-deal-monty-hall-problem/347123.md?page=4)