# Monty Hall problem on MythBusters **URL:** \ **Category:** Cafe Society\ **Created:** [November 26, 2011, 7:06am UTC](https://boards.straightdope.com/t/monty-hall-problem-on-mythbusters/604134 "2011-11-26T07:06:38Z")\ **Posts on this page:** 20\ **Page:** 3
**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:** [November 26, 2011, 6:42pm UTC](https://boards.straightdope.com/t/monty-hall-problem-on-mythbusters/604134/41 "2011-11-26T18:42:06Z")
> [@Nametag](#): > > I think people stick because they assume a far simpler mechanism is at work – Monty opens a door to try to get you to switch. He wouldn’t do that if you had picked the wrong door in the first place (except for a few to throw off suspicion). The only way to assess the accuracy of this hypothesis is to watch hundreds of games and determine whether picking the right door the first time is correlated with the reveal-and-offer. Thing is, knowing that’s what people would usually do means that any smart person would do the opposite. It seems odd that most people would think the scenario part of the way through, and not all the way. What’s also odd is the version of the test not shown: where one person repeats the test over and over, with there always being an offer to switch. Even then, it takes humans longer to figure it out than it does birds. Even when we know it’s random, we still stick to the hypothesis that they’re trying to trick us. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 26, 2011, 6:46pm UTC](https://boards.straightdope.com/t/monty-hall-problem-on-mythbusters/604134/42 "2011-11-26T18:46:59Z")
> [@Martian Bigfoot](#): > > There isn’t. It’s dead simple. Your cat could understand it. Maybe not cats, but [pigeons get it just fine](http://www.livescience.com/6150-pigeons-beat-humans-solving-monty-hall-problem.html). Y’all are smarter than pigeons, right? ---
**Author:** ![standingwave](https://avatars.discourse-cdn.com/v4/letter/s/9de0a6/32.png) [@standingwave](https://boards.straightdope.com/u/standingwave)\ **Post date:** [November 26, 2011, 7:01pm UTC](https://boards.straightdope.com/t/monty-hall-problem-on-mythbusters/604134/43 "2011-11-26T19:01:52Z")
> [@Quimby](#): > > The breakthrough for me that helped me understand the math is when I realized that by switching I am actually getting the other two doors. It’s just that I know for sure one of the two is definitely a loser (which is a given anyway). Yeah, and they actually did a good job explaining that when they went through the mathematics towards the end of the segment. ---
**Author:** ![OpalCat](https://avatars.discourse-cdn.com/v4/letter/o/8edcca/32.png) [@OpalCat](https://boards.straightdope.com/u/OpalCat)\ **Post date:** [November 26, 2011, 8:03pm UTC](https://boards.straightdope.com/t/monty-hall-problem-on-mythbusters/604134/44 "2011-11-26T20:03:55Z")
> [@Martian Bigfoot](#): > > For me, at least, the the problem is easier to visualize if it’s scaled up: Imagine a really rubbish game show where instead of three doors, there are 100 doors. You choose one, and then Monty says: “I’ll make this easy for you, because this is the worst game show ever. I’ll now open 98 of the remaining doors, where there is no prize. Only one of them remains closed. You can keep your door, or switch to my door instead.” > > Now, think. There are really only two sets of doors: The one door you picked, and the set of doors you didn’t pick. With 100 doors instead of three, your chance of picking the right door on your first pick was just one percent. The chance of the prize being behind a door \*somewhere \*in the set of the doors you didn’t pick, is 99 percent. > > When Monty asks if you want to switch, he’s really asking: “Do you want to keep the door that you initially picked, or switch to the full set of doors that you didn’t? I’ve already opened 98 of those doors, so if the prize was anywhere in this set from the get go, and you switch, you’ll win, guaranteed.” > > In other words, if you picked the right door to begin with, you win if you keep your door. If you picked the wrong door to begin with, you’ll win if you switch. But your chances of picking the right one at the start was only one percent! > > In this case, the answer should be a no-brainer. Doh, switch every time! You’ll get a 99 percent chance of winning! > > Now, in the actual Monty Hall problem, the situation is _exactly the same, just with different numbers._ Your chance of picking the right door to begin with was 1/3. The chance of picking the wrong one was 2/3. Switching gives you a 2/3 chance of winning. Great explanation! It makes it all make so much more sense! > [@Little\_Nemo](#): > > One thing I noticed was that Jamie said there was no point in telling us how many of the first group won or lost because they all picked to stay so their results wouldn’t compare the two strategies. But he was wrong. It’s a game with only two outcomes - so everyone who lost by staying would have won by switching (and vice versa if that had happened). So if we knew, for example, that seven people stayed and won and thirteen people stayed and lost, then we’d know that switching would have won in those other thirteen cases. That bothered me, too. Why wouldn’t those results be valid data? I asked my husband the same question at the time. ---
**Author:** ![Heller\_Highwater](https://avatars.discourse-cdn.com/v4/letter/h/278dde/32.png) [@Heller\_Highwater](https://boards.straightdope.com/u/Heller_Highwater)\ **Post date:** [November 26, 2011, 8:14pm UTC](https://boards.straightdope.com/t/monty-hall-problem-on-mythbusters/604134/45 "2011-11-26T20:14:37Z")
There’s no reason why it wouldn’t be. It’s just not the point of that particular experiment. They were only testing the “do people stay or switch” part of the myth. I suspect they just wanted a more visual demonstration of “why it’s better to switch”, which they provided nicely with the comparison between Adam and Jamie. ---
**Author:** ![AlsoNamedBort](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/alsonamedbort/32/3213_2.png) [@AlsoNamedBort](https://boards.straightdope.com/u/AlsoNamedBort)\ **Post date:** [November 26, 2011, 8:23pm UTC](https://boards.straightdope.com/t/monty-hall-problem-on-mythbusters/604134/46 "2011-11-26T20:23:03Z")
> [@Heller\_Highwater](#): > > There’s no reason why it wouldn’t be. It’s just not the point of that particular experiment. They were only testing the “do people stay or switch” part of the myth. I suspect they just wanted a more visual demonstration of “why it’s better to switch”, which they provided nicely with the comparison between Adam and Jamie. Yeah, the visual part of these experiments is part of what makes Mythbusters so successful. There have been plenty of urban legend shows in the past, none of which lasted very long because they just told you whether something was true or not. It’s plenty easy to just say “It’s better to switch,” “The airplane will take off,” or “The toilet will/will not blow up.” But having the visual of all those red squares really sells the myth. ---
**Author:** ![GuanoLad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/guanolad/32/42_2.png) [@GuanoLad](https://boards.straightdope.com/u/GuanoLad)\ **Post date:** [November 26, 2011, 9:41pm UTC](https://boards.straightdope.com/t/monty-hall-problem-on-mythbusters/604134/47 "2011-11-26T21:41:55Z")
> [@Left\_Hand\_of\_Dorkness](#): > > Let’s try it out; I’ll fill in your guess more-or-less randomly in step 2. > > 1. I’ve got my number, but I’m not telling you what it is. > 2. You guess that it’s 47. > 3. I tell you that it’s not 1-46, 48-72, or 74-100. It’s either 47 or 73. > 4. You can now switch your guess to 73, or you can stick with 47. > 5. If you stuck with 47, you’d be wrong. If you switched to 73, you’d be right. Okay, this explanation finally clarified things for me. However, it still makes my head hurt. ---
**Author:** ![Marley23](https://avatars.discourse-cdn.com/v4/letter/m/45deac/32.png) [@Marley23](https://boards.straightdope.com/u/Marley23)\ **Post date:** [November 26, 2011, 10:08pm UTC](https://boards.straightdope.com/t/monty-hall-problem-on-mythbusters/604134/48 "2011-11-26T22:08:12Z")
I thought they did a very good job with it, actually. I’ve read the Straight Dope column a couple of times and probably a few of our threads on the problem, but the answer never really sank in. The machine was a good demonstration - and of course when they finally got around to explaining the math it was extremely simple. ---
**Author:** ![Little\_Nemo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/little_nemo/32/3120_2.png) [@Little\_Nemo](https://boards.straightdope.com/u/Little_Nemo)\ **Post date:** [November 26, 2011, 11:22pm UTC](https://boards.straightdope.com/t/monty-hall-problem-on-mythbusters/604134/49 "2011-11-26T23:22:12Z")
> [@OpalCat](#): > > That bothered me, too. Why wouldn’t those results be valid data? I asked my husband the same question at the time. What he said was that they couldn’t compare the success rate of the staying strategy against the switching strategy because nobody chose the switching strategy so they had no data. So they ran the second experiment where Jamie used the staying strategy while Adam used the switching strategy. But their assumption was incorrect. Because the two strategies mirror each other, you can make the comparison even if you only have the results from one strategy. The success rate of one strategy will always be the complement of the other. ---
**Author:** ![OpalCat](https://avatars.discourse-cdn.com/v4/letter/o/8edcca/32.png) [@OpalCat](https://boards.straightdope.com/u/OpalCat)\ **Post date:** [November 26, 2011, 11:24pm UTC](https://boards.straightdope.com/t/monty-hall-problem-on-mythbusters/604134/50 "2011-11-26T23:24:42Z")
Exactly. That’s what I thought. Just look at how often they won by staying, and subtract that from 20 to see how often they would have won by switching. Simple. ---
**Author:** ![Left\_Hand\_of\_Dorkness](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/left_hand_of_dorkness/32/7156_2.png) [@Left\_Hand\_of\_Dorkness](https://boards.straightdope.com/u/Left_Hand_of_Dorkness)\ **Post date:** [November 27, 2011, 12:19am UTC](https://boards.straightdope.com/t/monty-hall-problem-on-mythbusters/604134/51 "2011-11-27T00:19:11Z")
For anyone else having trouble, here’s a data set. In order to get started, here’s what you have to do: 1. Create a list of 15 “games.” For each game, choose the letter A, B, or C. That letter represents the door the treasure is behind. 2. Compare your list of 15 games to my list of guesses in the spoiler boxes below. 3. For each game, eliminate a letter that is neither on my list nor on your list. That letter represents the door that you open and show me does not contain a prize. (For example, if, for game 7, you chose B for the prize, and I guessed A, you’d have to eliminate C. If for game 8, you chose B for the prize, and I guessed B, you could eliminate A or C.) 4. On your first run-through, understand that I’ll switch my choice after your elimination. (For example, if I chose A, and you eliminated C, I’ll switch my choice to B). 5. Give me a point for each correct answer. If you don’t agree that this models the Monty Hall problem, please explain why. Step 3 is the most difficult step to understand: keep in mind that Monty will always open a door (thereby eliminating it as a possible choice), and he’ll never open the door with a prize or the door the contestant initially chose. If you do agree, then set up your list of 15 games, and then compare them to the spoiler boxed guesses below as described in the rules above. [spoiler] 1. C 2. B 3. A 4. C 5. A 6. B 7. B 8. A 9. C 10. A 11. C 12. B 13. C 14. B 15. A[/SPOILER] If you want a control set of data, run everything again, only this time I won’t switch. For yet more sets of data, Use the set above, but start on a different turn number ---
**Author:** ![GoodOmens](https://avatars.discourse-cdn.com/v4/letter/g/d26b3c/32.png) [@GoodOmens](https://boards.straightdope.com/u/GoodOmens)\ **Post date:** [November 27, 2011, 3:42pm UTC](https://boards.straightdope.com/t/monty-hall-problem-on-mythbusters/604134/52 "2011-11-27T15:42:21Z")
When I explain this to (high school) students, who are often stuck on the “it must be a 50/50 chance” idea, I usually put it this way: imagine I’m in a boxing match with Mike Tyson. There are only two possible outcomes, he wins or I win. Those outcomes certainly are not equally likely. Of course, some of these little whippersnappers don’t know who Tyson is, so I sometimes try a different tactic. Almost all my students play some sort of sport. So, you’re on a soccer team. Either your team wins or the opposing team wins. If there’s a 50/50 chance, there would be no reason to practice, study plays, or work out. The whole point of those is that they affect the probability of winning. I also agree that it looked like Adam had more wins than I would expect, but the sample size was fairly small. If they had repeated things a few thousand more times, the win rates would be pretty close to the theoretical values. ---
**Author:** ![Little\_Nemo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/little_nemo/32/3120_2.png) [@Little\_Nemo](https://boards.straightdope.com/u/Little_Nemo)\ **Post date:** [November 27, 2011, 3:48pm UTC](https://boards.straightdope.com/t/monty-hall-problem-on-mythbusters/604134/53 "2011-11-27T15:48:10Z")
> [@Interrobang](#): > > When I explain this to (high school) students, who are often stuck on the “it must be a 50/50 chance” idea, I usually put it this way: imagine I’m in a boxing match with Mike Tyson. There are only two possible outcomes, he wins or I win. Those outcomes certainly are not equally likely. > > Of course, some of these little whippersnappers don’t know who Tyson is, so I sometimes try a different tactic. Almost all my students play some sort of sport. So, you’re on a soccer team. Either your team wins or the opposing team wins. If there’s a 50/50 chance, there would be no reason to practice, study plays, or work out. The whole point of those is that they affect the probability of winning. > > I also agree that it looked like Adam had more wins than I would expect, but the sample size was fairly small. If they had repeated things a few thousand more times, the win rates would be pretty close to the theoretical values. The problem is a lot of people believe in the 50/50 fallacy: “I figure you either win or lose with a lottery ticket so you got a 50/50 chance.” ---
**Author:** ![Bryan\_Ekers](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bryan_ekers/32/183_2.png) [@Bryan\_Ekers](https://boards.straightdope.com/u/Bryan_Ekers)\ **Post date:** [November 27, 2011, 4:02pm UTC](https://boards.straightdope.com/t/monty-hall-problem-on-mythbusters/604134/54 "2011-11-27T16:02:27Z")
> [@ultrafilter](#): > > Maybe not cats, but [pigeons get it just fine](http://www.livescience.com/6150-pigeons-beat-humans-solving-monty-hall-problem.html). Y’all are smarter than pigeons, right? Well, how _big_ of a pigeon? ---
**Author:** ![The\_Tooth](https://avatars.discourse-cdn.com/v4/letter/t/c67d28/32.png) [@The\_Tooth](https://boards.straightdope.com/u/The_Tooth)\ **Post date:** [November 27, 2011, 4:54pm UTC](https://boards.straightdope.com/t/monty-hall-problem-on-mythbusters/604134/55 "2011-11-27T16:54:46Z")
It’s always seemed pretty simple to me. There’s a 1/3 chance the prize is behind the door you picked. So there’s a 2/3 chance it’s behind one of the other doors. The prize certainly isn’t behind the door Monty opens, so there’s a 2/3 chance the prize is behind the remaining door. So you switch. I saw Alan Davies and Marcus du Sautoy stage it using toy cars and plastic farm animals. Alan lost. ---
**Author:** ![MaxTheVool](https://avatars.discourse-cdn.com/v4/letter/m/9fc29f/32.png) [@MaxTheVool](https://boards.straightdope.com/u/MaxTheVool)\ **Post date:** [November 27, 2011, 6:02pm UTC](https://boards.straightdope.com/t/monty-hall-problem-on-mythbusters/604134/56 "2011-11-27T18:02:47Z")
A subtle but very important point to remember is that all of this only applies if the game show host is always, no matter what door you pick, going to open and reveal another, empty, door. As long as he’s always doing that automatically, then it’s just a question of math. If, however, the host can choose from among various things to do after you make your initial choice: (a) open and reveal another door (b) give you a chance to double or nothing © offer to phone a friend (d) etc then the problem becomes one of psychology and so forth, and if you’re someone who’s purely math-oriented, then a sufficiently clever game show host might realize that, and might do the monty hall thing ONLY if you’ve in fact chosen right initially, knowing that you are thus going to assume that the puzzle math is relevant, and that he will thus fool you into switching. ---
**Author:** ![Don\_t\_Panic](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/don_t_panic/32/398_2.png) [@Don\_t\_Panic](https://boards.straightdope.com/u/Don_t_Panic)\ **Post date:** [November 27, 2011, 6:07pm UTC](https://boards.straightdope.com/t/monty-hall-problem-on-mythbusters/604134/57 "2011-11-27T18:07:42Z")
> [@MaxTheVool](#): > > A subtle but very important point to remember is that all of this only applies if the game show host is always, no matter what door you pick, going to open and reveal another, empty, door. As long as he’s always doing that automatically, then it’s just a question of math. Yes. Change the rules, and you have a different problem. Again, that’s why it’s so important that the conditions are stated specifically. The classical Monty Hall problem has the host always open an empty door. That’s when it becomes an interesting problem, an apparent “paradox”, and a lesson in how probability works, instead of… well, some random stuff involving doors. ---
**Author:** ![The\_Controvert](https://avatars.discourse-cdn.com/v4/letter/t/dec6dc/32.png) [@The\_Controvert](https://boards.straightdope.com/u/The_Controvert)\ **Post date:** [November 27, 2011, 7:52pm UTC](https://boards.straightdope.com/t/monty-hall-problem-on-mythbusters/604134/58 "2011-11-27T19:52:49Z")
In the episode, Adam won 38/49 (a 77.5% win rate) whereas Jamie won 11/49 or a 22.4% win rate. The expectation is for Adam to win 66.6% and Jamie to win 33.3%, but the sample size is small and I suspect their rig may not have been properly randomized. During the episode, they said they randomized the doors on the stage, but if they just “randomly” stuck the stickers on the long paper sheet, that might not have been as good as a true random number generator. ---
**Author:** ![garygnu](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/garygnu/32/2924_2.png) [@garygnu](https://boards.straightdope.com/u/garygnu)\ **Post date:** [November 27, 2011, 9:02pm UTC](https://boards.straightdope.com/t/monty-hall-problem-on-mythbusters/604134/59 "2011-11-27T21:02:52Z")
> [@Martian Bigfoot](#): > > Yes. Change the rules, and you have a different problem. Again, that’s why it’s so important that the conditions are stated specifically. The classical Monty Hall problem has the host always open an empty door. That’s when it becomes an interesting problem, an apparent “paradox”, and a lesson in how probability works, instead of… well, some random stuff involving doors. As far as the psychology of it, a contestant walking into a three-door choice set would not know that the host would open a door, or that the opened door is known to be empty. If the revealed door is random, then switching to win really is 50/50. You also could be shown right away that you lost. ---
**Author:** ![Don\_t\_Panic](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/don_t_panic/32/398_2.png) [@Don\_t\_Panic](https://boards.straightdope.com/u/Don_t_Panic)\ **Post date:** [November 27, 2011, 9:08pm UTC](https://boards.straightdope.com/t/monty-hall-problem-on-mythbusters/604134/60 "2011-11-27T21:08:58Z")
> [@garygnu](#): > > As far as the psychology of it, a contestant walking into a three-door choice set would not know that the host would open a door, or that the opened door is known to be empty. If the revealed door is random, then switching to win really is 50/50. You also could be shown right away that you lost. Sure. So what? The host \*could \*also suddenly start farting out golden turkeys, or the door could be a magic portal leading to Narnia. But that’s not the Monty Hall problem. Don’t fight the hypothetical. [Previous page](https://boards.straightdope.com/t/monty-hall-problem-on-mythbusters/604134.md?page=2) [Next page](https://boards.straightdope.com/t/monty-hall-problem-on-mythbusters/604134.md?page=4)