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The Time Everyone “Corrected” the World’s Smartest Woman (2015)
- murat124 5y agoExcuse my GIMP skills. https://i.imgur.com/MW90pVZ.jpg https://i.imgur.com/MW90pVZ.jpg
- 1123581321 5y agoWhich public intellectual introduced Monty Hall to the public when awareness of the problem was at a similar low, and received the fewest incorrect responses?
- dllthomas 5y agoFor the glib answer: probably the public intellectual who received the fewest responses, period.
- 1123581321 5y agoThat's probably true, since satisfaction with an explanation tends not to motivate letter-writing in any situation.
- swayvil 5y agoIs "Vos Savant" her real name? It's kind of a giveaway.
- crooked-v 5y agoIt actually is the family name on her mother's side, though she changed it to that back from her mother's married name (Mach).
- BelenusMordred 5y ago> Mach Amusing that her alternate name still exudes intellect. https://en.wikipedia.org/wiki/Ernst_Mach https://en.wikipedia.org/wiki/Ernst_Mach
- GavinMcG 5y agoHer mother was Marina vos Savant.
- deleted 5y ago[deleted]
- smt88 5y agoVos Savant is her mother's maiden name, which she chose to use over her father's.
- the__alchemist 5y agoThe hubris of some of the her critics - especially ones with titles - is remarkable.
- psychometry 5y agoIt won't be long before we see some here I predict.
- devilduck 5y agoHappens here all the time
- pmiller2 5y agoI'm willing to forgive pretty much anyone who's fooled by something that also fooled Paul Erdos: https://www.wired.com/2014/11/monty-hall-erdos-limited-minds/ https://www.wired.com/2014/11/monty-hall-erdos-limited-minds...
- macintux 5y agoBeing wrong isn’t a crime. Harshly lambasting a woman for, largely, being a woman and disagreeing with you is the problem.
- nextlevelwizard 5y agoJumping to sexist assumptions is also a problem.
- azinman2 5y agoI guess it shows for those who wrote her about female logic that it was, in fact, they who were overly swayed by male logic.
- saeranv 5y agoI guess Monte Carlo simulations weren't well-known or easily accessible back then? It would have been the perfect way to shut up her critics.
- gameswithgo 5y agoi was about 15 when this went down and immediately wrote one that confirmed she was right. formative experience in my life.
- naikrovek 5y agothey were. the critics didn't use them. she told them to "try it" in one or two columns before she had to explain it very slowly and methodically in a third, as I recall. I'm old. I read each of these as they arrived in the paper.
- mcphage 5y ago> I'm old. I read each of these as they arrived in the paper. So back in the day, Parade magazine was a thing that people actually read, and not just ignored or recycled immediately? The past really is a foreign country.
- pmiller2 5y agoYou could do it in Excel in probably 20 minutes.
- 5y ago
- jim010101020202 5y agoI may be missing something (not a mathematician, probabilist) but doesn’t the fact that the host would never reveal the car change the calculation? In the table shown in the article game 3 & 6 are nonsensical so with the remaining 4 games the odds are even between switch & stay.
- andrewclunn 5y agoThis is the missed portion of the explanation. If one were watching the show for the first time, and unaware that the host ALWAYS revealed a goat, and that the door that they revealed was not random, then they could reasonably think that it is a 1/2 coin flip decision. This is almost always statisticians doing a poor job of explaining the problem, and then thinking themselves clever when people "get it wrong."
- crooked-v 5y agoThe table is for the host revealing one door that you didn't pick and that has a goat behind it, not for the host revealing door #3.
- newacct583 5y agoNo, it's correct. The easiest explanation, IMHO, is: 1. If you already selected the correct door, you should not switch, obviously. Equally obviously, the odds of being in this state are 1/3, because your initial choice was random. 2. If you did not select the right door, then the host clearly revealed the only one of the two remaining doors without a prize. In this state, you clearly should switch to the only door remaining, which contains the prize. That is, the likelihood that switching will get you the prize is 2/3, because it corresponds to the states where you initially guessed wrong.
- dionidium 5y agoThis is a very good explanation. I hope you won't mind if I say the exact same thing you did in just a slightly different way that might help it click for someone still not getting it. Your initial chances of choosing correctly are 1 in 3. So the car is behind the door you chose 1/3 of the time. After the host reveals which of the 3 doors definitely does not have the car it's still the case that your initial choice will have the car 1/3 of the time [0]. That means the other door must contain the car 2/3 of the time. So you should switch. [0] The haters are correct that the host hasn't changed the odds that your initial choice is correct. They just aren't tracking the implications of that fact.
- 6gvONxR4sf7o 5y agoThere are a number of things I like about the monty hall problem. There's the history, the unintuitiveness, the subtle easy-to-screw-up nature of probability problems, the calculation, the sociology, and the overconfidence of wrong experts. Most of all, it's the calculation vs intuition that I like. You can do the calculation, or run simulations and prove correctness. In fact, it would be much harder to be so widely wrong now that any statistician can so easily just code up a computer to run it a zillion times and get an empirical answer. But despite that, your intuition might still say otherwise. I'm fascinated by the gap between intuition and logic particularly because it can be closed as soon as you find the right lens to take to a problem. I like the example in the article about 100 doors, or even a million doors, where it really helps drive it home in your gut. That moment when you find the right intuition for a thing is a magical one that makes me love math.
- darig 5y agoWhen I first heard of it in high school, I immediately coded a simulation and had an answer in less than 5 minutes that I could be confident about. I would have sat second-guessing myself for 5 minutes had I not done that.
- paulcole 5y agoMy favorite thing is when somebody says, “what if I want to win the goat?” Because you instantly know they’re a contrarian for its own sake.
- deleted 5y ago[deleted]
- Spivak 5y agoI don’t think you need the 100 or 1,000,000 doors thing. If you decide before not to switch then your goal is to pick the car which you have a 1/3 chance to do. If you decide before to switch then your goal is to pick a goat which you have a 2/3 chance to do.
- deleted 5y ago[deleted]
- calibas 5y ago"You made a mistake, but look at the positive side. If all those Ph.D.’s were wrong, the country would be in some very serious trouble."
- inopinatus 5y agoAnd that, boys and girls, is how we got adtech.
- fastball 5y agoThis is why I no longer call people Dr.
- failwhaleshark 5y agoLol. I worked at a biomedical informatics shop with IITians and other folks who were MD + PhD CS. No one referred to anyone as doctor and no one there took a MENSA test. I think it's safe to say the only "doctors" are clinical MDs, podunk colleges to seem elitist, and Spies Like Us during the surgery scene.
- leephillips 5y agoOnly losers take Mensa tests and talk about their IQs. But plenty of PhDs use the Dr title in the right context. It's just the height of jerkdom to insist on people addressing you with it.
- listenallyall 5y ago> the height of jerkdom to insist on people addressing you with it Tell that to Dr. Jill Biden. Who doesn't even have a Ph.D.
- fastball 5y agoWhat's the right context? Why are PhDs (and MDs) the only people that deserve a special prefix in our society? I know loads of people that have worked harder and smarter than people with PhDs, but they don't get a prefix.
- jml7c5 5y agoFrom reading the problem as posed in the article, the confusion seems similar to the "plane on a treadmill" thing, where people are interpreting the premise of the problem differently. (In plane on a treadmill: "the treadmill moves backwards @ -1 * plane airspeed" vs "the treadmill moves backwards to keep plane airspeed at 0". In this: "the host will open the door at random and in this example it happen to have a goat" vs "the host will never open a door with a car behind it".)
- dllthomas 5y agoThe presentation of the problem included the phrase "the host, who knows what's behind the doors, opens another door", so I think in that case the interpretation that Monty deliberately reveals a goat is more appropriate (in addition, obviously, to being what was intended), although I agree there is wiggle room enough to admit both possibilities and clarity about which case you are discussing is important as it changes the answer.
- jVinc 5y ago> as it changes the answer. It doesn't. If the host randomly reveals a car, then you have 0% chance to win. If he doesn't you have 66% chance to win by switching.
- dllthomas 5y agoI once thought that had to be the case. I wrote the simulation to demonstrate that I was correct. I learned that I wasn't. I encourage you to do likewise. I think I now have an understanding of why the right answer is the right answer, but I thought I had that before; I am more confident in the answer than my reasoning.
- deleted 5y ago[deleted]
- jVinc 5y agoI owe you an apology, you're correct, I was wrong. I can't pinpoint where my reasoning failed me exactly, but my confidence certainly was misplaced, and that's from someone who did do the simulation originally to understand the outcomes, but not to understand the distinction between the host deliberately or randomly revealing the door. I'm sorry, but thank you for you persistence in correcting me. It was a fruitful learning experience to dig into this problem again.
- kingsuper20 5y agoI'm familiar with the problem and my intuition is still wrong. I wonder if you couldn't design a gambling machine that has a variant of the Monty Hall problem built-in favoring the house (naturally).
- crooked-v 5y agoFrom what I understand, that would just be overcomplicating the matter, because most modern video gambling machines just use a set number of wins per time interval.
- trothamel 5y agoWhat's more, laws tend to limit how much the machines pay out, not how little. (Other than making sure the game is 'fair'.) Someone who loses a lot tends to limit how much they have to play. By contrast, the thrill of frequent winning gets people to lose more, in total, then they otherwise would haver.
- tebou 5y agoThat is incorrect, and most likely illegal anywhere gambling is legalized. The machines are random, with no enforcement on wins per time interval. Source: I work in the industry.
- deleted 5y ago[deleted]
- freeopinion 5y agoI have a Ph.D. in slot machines and you blew it big time. In Nevada they are very much controlled, just not by time interval. Oh, and fwiw, I know several people who always win when they come to Las Vegas. I know because they've told me so. This proves that you don't actually work in the industry. (I really hate that I have to disclose that I don't really have a Ph.D. is slot machines.)
- kingsuper20 5y agoActually they use a RNG that in the case of slots, has a kind of virtual reel with multiple assignment. If you like, add a bit of skill that gives the primates the feeling of control (video poker) or when you let loose of the handle or press a button (which simply grabs a number). Stir in a bit of marketing pizzazz (off-by-one on slots, enormous number of simultaneous smaller bets, low payoff better odds vs. high payoff worse odds) and you can get the mark to stand there for a bit longer. Networking, recognition and databases on players, etc. add more wrinkles. In any case, I don't doubt that a Monty Hall bet could be wrappered into the game somehow to give the appearance of better odds than really exist. (hopefully I got this somewhat right, I've never designed any gaming stuff. I'll bet that a gig at Bally or IGT would be a real eye-opener.)
- javajosh 5y agoSo this is at least in part about the "Monty Hall Problem" and why it's solution not intuitive. The article missed an important angle: when the host opens a door, he's giving you more information, which explains why it's better to switch. If you're the host, you need to know which door the car is behind to do your job 2/3 of the time, to avoid revealing it. It's this quality of unexpected information exchange that I find most fascinating about this particular puzzle!
- dllthomas 5y agoInterestingly, if the host picks randomly (and if he reveals the car, you... start over, or you get the car, or you get nothing, or ... it doesn't matter because it happens not to have happened in the time we're considering) then you are faced with a 50/50 chance.
- TchoBeer 5y agoI don't understand how this could be. If you start over when the host picks the car, then isn't that the same as the host picking the goat every time, i.e. the same as the host knowing.
- inopinatus 5y agoWell, I don't know. Advantage moves to the contestant: they close the doors, but everything's gonna remain in the same place. Steathily repositioning an automobile on a soundstage might be possible (albeit difficult), but just you try moving the damn goat
- deleted 5y ago[deleted]
- HALtheWise 5y agoFair warning: I haven't rigorously verified the math on this, and tfa is all about the importance of vigorously verifying the math. In the case where the host revealing the car means you restart, there are three equally likely situations after the host opens a door: - You picked the car, and the host revealed a goat - You picked a goat, and the host revealed the other goat - You picked a goat, and the host reveals the car (restart) Of the two terminal cases, one gets you the car if you switch, and one gets you the car if you don't switch. In the non-terminal case, it doesn't matter what policy you have because you don't even get a chance to apply it.
- dfxm12 5y agoHall clarified that things worked a bit differently than the scenario presented by the Parade reader in vos Savant’s column. This is partly why I think it's hard for some to comprehend "the Monty Hall problem". If you've ever seen Let's Make a Deal, you'll notice the big deal is set up very close to, but still materially different than how "the Monty Hall" problem is set up. I wonder if people who have never seen the show can understand the math quicker...
- Cpoll 5y agoAnother statistically unintuitive problem (which I've witnessed a lecture hall enter a state of uproar over): There are 2 red and 2 blue balls in a box. One ball is removed at random, what are the odds that the ball is blue? Now we repeat the problem, but before examining the ball, we remove a second ball. We observe that the second ball is blue. In this case, what are the odds that the first ball is blue?
- paxys 5y agoProbability of draws: B then R -> 1/3 B then B -> 1/6 R then B -> 1/3 R then R -> 1/6 Removing the cases where the second draw was R: B then B -> 1/3 R then B -> 2/3 Therefore the first ball being blue has a 1/3 chance.
- deleted 5y ago[deleted]
- norswap 5y ago1/3 - If the first ball was blue, the probability to pick the second one is 1/3, while if it was red, the probability to pick a blue ball as second ball is 2/3. The prior probability for each scenario (first blue ball vs first red ball) is 1/2, so the posterior is 1/3 that the first ball was blue. A neat trick to reason about those cases is to use odds. The prior odds for the first ball being blue vs red is 1:1. The odds for the first ball being blue vs red, given the second ball is blue is 1:2. We can just multiple the odds to get (1*1):(1*2) = 1:2 as the posterior odds. This doesn't seem impressive because the prior is 1:1, but using this method you can easily calculate the odds in the scenario where there are 4 red and 2 blue balls. The prior odds is 2:4=1:2, the conditional odds is (1/5):(2/5)=1:2, 1:2 * 1:2 = 1:4, i.e. 1/5 chance that the first ball was blue.
- cableshaft 5y agoYeah, 1/3. Once I labeled them as R1 R2 B1 B2 and just listed all possibilities, it became clear where my error was.
- amscanne 5y agoLove it. The answer is 1/2. The first event occurs with a population of 2,2 and the second event has no causal effect. The trap is that the observation doesn’t change the sample population of the first event, exactly like the Monty Hall case: opening doors itself does not change the probability to 1/2 as people think it does (it remains 1/3) — it’s the switch that changes the odds. Similar question with a similar trap: there are new neighbors moving in next door, and you know they have two kids. You see a boy in their yard, so you know they have at least one boy. What’s the probability they have two boys?
- deleted 5y ago[deleted]
- randyrand 5y agoAn intuitive way to think about the problem is from the perspective of the host. Sometimes the host has 2 goats he can pick from to show, and sometimes he only has 1 goat he can show. 66% of the time the contestant chose a goat, so the host has to reveal the only other goat 66% of the time. So switch.
- hliyan 5y agoThis is the most intuitive formulation for me.
- version_five 5y agoThe way I think of it, if you pick a goat on the first try (2/3 chance), you get the car by switching
- graderjs 5y agoThe 2 goats didn't make sense to me but the host perspective does I think, like: after player's first choice, there's two cases. Either player picked the car, or they didn't. If player picked the car, then host has two choices to pick from, if player didn't pick the car, the host has 1 choice to pick from. (At this point you can sort of imagine a binary tree with 3 leaves and 5 nodes) So either player picked the car the first time with a 1/3 chance (so, less likely), or didn't (more likely). So it's more likely for player to miss the first time (unless you're a precog psychic like me), which, "paradoxically (but not really)", by missing (by failing), player actually improves odds for second guess. So, on average the second guess is more likely to hit, because, on average, the first guess is more likely to miss. Now finally it makes sense to me. Because it's more likely to miss on first guess, it's also more likely to hit on second guess, so it's more profitable to switch, on average. Do I get it now? I think I do! Woot! :P :) xx
- umvi 5y agoIf I were her I would have used the opportunity to get rich. "Note to the 10,000 people who sent me letters saying I'm wrong: I'm willing to put my money where my mouth is, are you? I will pay anyone that can experimentally prove I am wrong $1000. If you can't do it, you owe me $1000. You have to put up the money before attempting any experiments. Put up or shut up."
- CyberRabbi 5y agoSmart lady but IQ tests are definitely not useless, if you disagree try this test: https://test.mensa.no https://test.mensa.no
- mcphage 5y agoWhy would talking that test convince me of the usefulness of IQ tests?
- kbelder 5y agoHow they're useful is debatable. But if you had a hundred people take an IQ test, and divided them into quartiles, you would notice some clear and strong differences between the groups, in a lot of different aspects. What to do about the differences is an exercise left for the reader.
- deleted 5y ago[deleted]
- robocat 5y agoIQ tests apparently have a very wide standard deviation, and regression to the mean would also be an issue (if you select for high IQ, you are also selecting for medium IQ with large positive error).
- froh 5y ago> Whereas only 8% of readers had previously believed her logic to be true, this number had risen to 56% by the end of 1992, writes vos Savant; among academics, 35% initial support rose to 71%. that 71% leaves 29% of _academics_ not getting the elementary math of the problem. I'm baffled this is _that_ hard?
- paxys 5y ago"Academics" is a very loose definition. I imagine a very low percentage of PhDs out there have studied math seriously beyond a middle school level.
- irrational 5y agoThat 29% is probably made up of men that honestly think she should be pumping out babies instead of doing men's work. I wish I was being sarcastic or otherwise kidding.
- antonvs 5y agoI think the lesson is that the are a lot of bad academics out there, who either repeat what they've been told without understanding it themselves, or whose confidence exceeds their ability, Dunning-Kruger style.
- failwhaleshark 5y agoThis reminds me of the time my entire family screamed and shouted that I was wrong that buying two different lottery tickets slightly more than doubles the total (infinitesimal) odds of winning over just one. 1 in a zillion vs slightly greater than 2 in a zillion because eliminating one choice reduces the pool by one for the next choice.
- deleted 5y ago[deleted]
- leephillips 5y agoIf you are talking about a simple lottery with N tickets and a fixed probability of any one ticket being the winner = 1/N, then the screamers were correct. To see this, imagine that there are a total of two tickets (N = 2). You buy one ticket. The probability that it is the winner = 1/2. After buying the second ticket, is your probability of winning somehow > 1? Or has it exactly doubled?
- deleted 5y ago[deleted]
- qot 5y agoBut you are wrong: it's exactly double the odds. If you bought all the tickets, do you think you'd have over 100% chance of winning? Edit: looks like leephillips beat me to it
- failwhaleshark 5y agoMea culpa. Instead of arguing with the shifting sands of ambiguous and imprecise English, it's better to use equations and evidence. P(A) = first ticket wins P(B) = second ticket wins P(A|B) + P(A|^B) + P(^A|B) = 1 - P(^A|^B) = 1 - (1 - 1/N) * (1 - 1/(N-1)) = 2 / N It is slightly > 2 / N if the ticket is revealed before the next one is chosen because then it does throw away that possibility.
- aaplok 5y ago
- paulpauper 5y agogood illustration https://upload.wikimedia.org/wikipedia/en/thumb/c/cd/MontyInNutshell_Draft.png/640px-MontyInNutshell_Draft.png https://upload.wikimedia.org/wikipedia/en/thumb/c/cd/MontyIn...
- FabHK 5y agoOk, so this is about the Monty Hall problem. A few remarks: 1. It is crucial exactly how the problem is framed. The "correct" framing is: a) Contestant choses a door, b) Monty Hall will pick a door (different from the one the contestant chose) SUCH THAT a goat is behind it (if there are two, pick one randomly), c) contestant is offered a choice to switch. In that case, the correct answer is to switch, increasing the chance for a car from 1/3 to 2/3. (The alternative framing is this: a) Contestant choses a door, b) Monty Hall picks a door randomly (different from the one the contestant chose), and if a car is behind that door, then the game ends, otherwise, c) contestant is offered a choice to switch. In this case, switching doesn't matter, the chance of getting the car is indeed 50/50, switching or not.) 2. Not everyone "corrected" her, some people did, and some people supported her. The confusion arose, arguably, because of the unclear framing. (One could argue that the phrase "the host, who knows what’s behind the doors" hints that the first framing is intended, but why not make it explicit? The host could know what is where and still choose randomly, in which case we're at the alternative framing.)
- phamilton 5y agoThe simplest explanation I've heard that has helped it click for people is "You can choose Door A, or both Doors B and C". Clearly choosing both B and C is a 2/3 probability. Then you guide them to understand how the host revealing one of B or C doesn't detract from the fact that you are still choosing both B and C to begin with. And finally establishing that switching is equivalent to choosing B and C instead of A.
- mariodiana 5y agoI read about this years ago in a very good book (for the layman) on the subject of probability: "The Drunkard's Walk." I highly recommend the book.
- version_five 5y agoOne appealing thing to me about math is how egalitarian it is. It doesn't matter where ideas come from, they are probably right or wrong. It is (maybe unhealthily) rewarding to see critics acting like its another discipline where the senior or otherwise higher status person's opinion has more weight, only to make themselves look ridiculous. Contrast this with so many workplace things where senior people say dumb or incorrect stuff all the time but there is not the same objective way to call them out on it.
- elliekelly 5y ago> One appealing thing to me about math is how egalitarian it is. I simply cannot believe anyone could read this article and come to this conclusion. Do you think the same flood of unduly harsh criticism would have resulted from the same column written by Maury von Savant instead of Mary?
- tmule 5y agoThere’s a bit of historical revisionism at play in the article. Here’s an accurate representation by Aaron Brown of what transpired: https://www.quora.com/Why-do-some-PhDs-argue-against-Marilyn-Vos-Savant-s-conclusion-on-the-Monty-Hall-problem-https-youtu-be-4Lb-6rxZxx0 https://www.quora.com/Why-do-some-PhDs-argue-against-Marilyn... Quoting for those who don’t want to go off-site: “You have to have lived through this ancient travesty to care about it. It’s probably best forgotten. Marilyn Vos Savant published an incorrect answer to the Monte Hall problem. She got the correct answer from a lot of readers, including me. She then published a second incorrect answer claiming that all the letters she got—including from math PhDs were wrong. Finally she got the answer right—the one she undoubtedly got from any math PhD or intelligent person who wrote her—but continued to claim (a) that her previous answers were correct and (b) that all the letters were wrong. As a result, the public is mostly misinformed about this simple problem. The set-up is simple. There are three doors, one of which has a valuable prize behind it, and two of which have worthless prizes. You select one door. The host opens one of the other two doors to show you a worthless prize, and offers you the opportunity to switch your choice. The key to this problem, which Marilyn finally saw in her third solution, is the knowledge and intentions of the host. If the host doesn’t know which door has the prize, there’s no reason to switch or not switch, it’s the same either way. If the host knows the door and is trying to hurt you, don’t switch. If the host knows the door and is trying to help you, switch. Marilyn’s first answer was to always switch, regardless of the host’s intentions. This is also the answer that most people now believe. It’s a common error that people make all the time. That’s why people wrote in to correct her. In order to justify her incorrect first answer, she claimed it was obvious that the host used the strategy of always opening a non-prize door and offering you the chance to switch. This does indeed justify switching. But that assumption was missing from her first and second answers, along with any discussion of the host’s knowledge and intentions mattering. Her justification that the assumptions were too obvious to need stating was the actual television program Let’s Make a Deal, hosted by Monte Hall, which eventually gave the name to the problem. But anyone who watched the show knew that was not Monte’s strategy. Sometimes he opened another door and gave a chance to switch, sometimes he didn’t. He was well aware of the problem if he followed Marilyn’s assumed strategy, and was scrupulous about not giving any advantage to the contestant. You can tell this from statistics on the show—on average it didn’t pay to switch—and his published statements. Clearly there are much bigger problems in the world, and it doesn’t pay to get upset about misinformation put out by popular vain people. If was very frustrating at the time, but it seems quaint that such things upset us in the 90s when there is so much worse information put out today by even more popular and more vain people.“
- hellbannedguy 5y agoI still don't get it. 1. Three doors. 2. Two doors have a goat, and one has a car. 3. The contestant has a 1/3 chance of winning the car? 4. The contestant loses. 5. New game, and odds? 6. There's a 50/50 chance of winning? (I'm assuming Monte Hall has no clue to where the car is. He is just opening doors.) 7. Could someone explain it to me, and thanks in advance. (Off topic but a fawn had two babes in my back yard. It was pretty amazing. At first, it looked like the mother abandoned her babies, but she didn't. The babies stayed in the same spot for two day. By the third day, I thought she abandoned them. I looked into care. I was informed to leave them alone. They said, if you knew for sure the mom was killed, you could bottle feed them Goat's milk. Cow's milk is not good for baby deer. I didn't overreact, and mom was there all along. She now just visits with her healthy kids. This has been my best spring in memory. Happy Mother's day to the moms out there.)
- listenallyall 5y agoMonte Hall DOES have a clue where the car is, and is NOT "just opening doors" With the full knowledge of where the car is, Mr Hall opens a door that is NOT the door the contestant chose, and is also NOT the door containing the car.
- the__alchemist 5y agoThis is the crux of it!
- atdrummond 5y agoYou have three doors. As you note, you have a 1/3 chance of winning. Monty now reveals a losing door. At this point, your door has a 1/3 chance of still winning - the probability of that choice can’t change. However, as we now know one door has a 0/3 chance of winning (it’s been revealed) the remaining door must have (1-1/3) chance of winning. Thus, the remaining door has a 2/3 chance of winning.
- listenallyall 5y agoI'm going to disagree with your framing here. Probabilities DO change as more information becomes available. For example, prior to rolling a pair of dice, you have a 1/36 chance of rolling a 12. However as sooner as you see the value of the first die, the probability changes to either 0 (most likely) or 1/6. In your description, if Monte opens a door randomly -- which you don't make clear whether he does or doesn't -- then the presence of the goat DOES change your P(winning a car).
- dantillberg 5y agoI find that the phrasing of the Monty Hall problem makes the solution less certain than stated. > "You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat." That's what the gameshow host did in this one trial, but why do we assume that the host would do the same thing every time? Perhaps the host only tries this other-door-goat diversion tactic when the contestant first selected the door with the car!
- mikewarot 5y agoWhat does it feel like when you're wrong about something? It doesn't... it only feels bad after you figure out you've been wrong. I love sharing this puzzle/problem, and hope to once again share it with new friends once the plague is over.
- bnkamalesh 5y agoI'm no way an expert, so even if Ms. Savant beautifully explained it; my puny brain thinks... "how is it different from starting the game with 2 doors?". If you choose again, how is the probability not 1/2
- stordoff 5y agoBecause you aren't starting from scratch - that you had a 1/3 chance to begin with is important. Say there are three doors, A B and C. You pick A: P(A wins) = 1/3 P(A loses) = P(B _or_ C wins) = 2/3 The host then reveals that there was a goat behind door B. This doesn't change the state of anything (there is always a losing door in the two you didn't pick, and he is always choosing to show that one, and none of the items move). This means the probabilities remain as they were above. However, we know that B didn't win, so we can simplify it to: P(A wins) = 1/3 P(A loses) = P(C wins) = 2/3 Therefore, if you switch to door C, you have a 2/3 chance of winning, rather than 1/3. The only way the probabilities would go back to 1/2 for the second choice is if the prize and the goat were shuffled after B is revealed. However, they are not, so the chance that you picked the right door initially is fixed when you picked it. To think about it another way, your initial choice of A means there's a 1/3 chance it is in A, and a 2/3 chance it is not and is behind one of the other doors. By ruling out B, we don't change the 1/3 chance that it was initially behind A. That means when we are asked again, there is still a 1/3 chance it was behind A, and a 2/3 chance that it was not. However, there's now only one thing that is not A, so there is a 2/3 chance it is behind door C.
- megablast 5y agoBecause you are throwing away information that you already have.
- utopcell 5y agoYou pick the car with 1/3 probability and a goat with 2/3. If you picked the car and switch, you lost. If you picked a goat and switch, you win because the only other door left has the car. The problem is simple to reason through. The hard part is convincing yourself that you need to think through it given what seems "obviously correct".
- h0l0cube 5y agoThis is the same explanation I arrived at too. Crucial to answering the problem correctly is understanding that the host doesn't reveal a random door, but a door that you have not chosen, and also is wrong. And then the reasoning follows naturally to the correct answer. It's amazing to be so prejudiced that someone would be adamant on the incorrect answer, when it can be explained in 3 simple lemmas.
- asciimov 5y agoReading "Ask Marilyn" in the paper was one of those Sunday rituals that I miss from my childhood. Along with the puzzler from Will Shortz on NPR on the way to church and reading the Sunday funnies. I kinda miss the newspaper, the way it was back then.
- elliekelly 5y agoI’m disappointed that this entire thread is focused only on discussing the Monty Hall problem and conveniently pretends her gender, and how she was treated, is unworthy of discussion. Come on! If we can’t even mention how STEM women are often treated as default-incompetent in this, _very_ egregious case then we have no hope of being able to have the discussion at all.
- Mountain_Skies 5y agoInteresting that they chose to identify Marilyn vos Savant in the headline as "Woman" instead of by her name. Also interesting that in doing so, they qualified her then believed to be world's highest IQ, male or female. Looks like they were attempting to be woke but ended up being misogynistic instead. It's also worth noting that many people did agree with her so it wasn't "everyone" correcting her.
- fisherjeff 5y agoAny article that is secretly about the Monty Hall problem should have a very prominent trigger warning at the top
- dec0dedab0de 5y agoThe problem with the Monty Hall problem is that it's based on implicit rules. Every explanation I've ever hear never says that no matter what they will never remove the winning choice.
- inopinatus 5y agoThere is a written rule: it's illegal. The Communications Act was amended in 1960 making it a federal offense for game shows to rig the outcome. Surreptitiously and intentionally removing the main prize on offer easily qualifies as just such a deceptive practice. https://www.visitthecapitol.gov/exhibitions/artifact/communications-act-amendments-september-13-1960 https://www.visitthecapitol.gov/exhibitions/artifact/communi... if you need that in ink & vellum. Unlawful behavior can be reasonably excluded from both the construction and the answers to logic puzzles, except when noted otherwise (or otherwise reasonable in context), which is why taking Monty hostage, busting through all three doors, jacking the car, snatching the goat, and driving off into the sunset, is not a reasonable response to the problem either, no matter how great TV it might make for.
- megablast 5y agoThey show all the doors. Why would they ask you to switch doors, if they already shown you the car?? Think about it.
- dec0dedab0de 5y agoAhh yeah that does make sense. I didn't realize they actually show you what the prize would have been.
- b0afc375b5 5y agoMy intuition keeps betraying me, but a truth table finally convinced me: | Car | Goat | Goat | Switch? | Win? | |-----+------+------+---------+------| | T | F | F | F | T | | T | F | F | T | F | | F | T | F | F | F | | F | T | F | T | T | | F | F | T | F | F | | F | F | T | T | T | Number of wins when not switching: 1 Number of wins when switching: 2 EDIT: Woops, hackernews does not like an org-mode table. Found out you can indent by 2 spaces to preserve format.
- snitzr 5y agoI wanted to prove this to myself programmatically so I made this Python script: https://gist.github.com/snitzr/7ecf0abaacce610893e9364b13bce205 https://gist.github.com/snitzr/7ecf0abaacce610893e9364b13bce... It’s better to switch. One way to think about it is in an exaggerated way. If there was a googolplex number of choices and you chose one and then all but your choice and another door were taken away, you’d have almost a 100% chance of being correct if you switch.
- hoten 5y ago“The winning odds of 1/3 on the first choice can’t go up to 1/2 just because the host opens a losing door,” Wow. I've never seen it phrased so simply. _of course_ it doesn't change–if you picked the door with a goat, the host has no choice but to open the door he does. If you pick the door with the prize, whatever door he picks simply doesn't matter. There's no new information to learn about your own pick from that.
- deleted 5y ago[deleted]
- randompwd 5y ago2016
- dudeinjapan 5y agoThe problem stated as stated by vos Savant hinges on the words "Monty (the host) knows". It implied that Monty intended to only open a door with a goat, so using his behind-the-scenes knowledge he opened door #2 (goat). However, such knowledge does not imply intention a priori. It could be that Monty doesn't give a f---, and he made the choice completely at random, for example by flipping a coin. Under this interpretation, he could have chosen door #3 (car) with equal likelihood, despite knowing that it would end the game early. In this case, the 50-50 answer would be correct. I think this is what trips many readers up, given that we tend to associate impartiality with game show hosts. A better statement of the problem might be something like "The producers have instructed Monty in advance that after the initial selection, he must open one of the unselected doors to reveal where a goat is (under such instruction he will never open the door with the car.)"
- Tarsul 5y agojust the fact that we have SO many comments here (and always when it comes to this topic) about how this problem can be misunderstood (e.g. knowing that the showmaster will never open the winning door before yours and that he will ALWAYS show you a blank door), means that we don't quite know how bad people are in intuitively understanding the probabilites because the real problem here (and elsewhere) was the meaning of the question, not (only) finding the answer. Nonetheless, we have a lot of studies that show how bad we are with probabilites (e.g. that the probability of two defined rare things happening at the same time is not only rare but extremely rare - unless they are interconnected of course) so there is certainly truth in here, but not quite as much as it appears at first. Nonetheless, I really like how much more intuitively clear the problem gets once you're talking about 100 doors instead of 3: So if the question were: "You're taking 1 out of 100 doors, say number 27, and then the rules of the game say that the showmaster must open 98 of the other doors that all hold a goat, meaning only your door and another are remaining. In one of those is certainly the price. Will it be rather in yours or in the other? Remember that you chose randomly out of 100 doors while the shomaster had to take care not to take out the winning door when he took out the other 98 doors (unless it was already your number 27, then he could have chosen freely). So where is probably the price?"
- js2 5y ago> In Joseph Bertrand’s box paradox (1889), three boxes are presented -- one containing two gold coins, one containing two silver coins, and the final containing one of each. Assuming the participant draws one gold coin from a box, the problem then asks what the probability is that the other coin in that box is gold. Bertrand, who concluded that the probability was ⅔, was lauded for his ability to look beyond the obvious. The initial distribution is: [gold, gold] [gold, silver] [silver, silver] With six possibilities for drawing a coin: 1. [----, gold] [gold, silver] [silver, silver] 2. [gold, ----] [gold, silver] [silver, silver] 3. [gold, gold] [----, silver] [silver, silver] 4. [gold, gold] [gold, ------] [silver, silver] 5. [gold, gold] [gold, silver] [------, silver] 6. [gold, gold] [gold, silver] [silver, ------] After drawing gold, you are left with: 1. [----, gold] [gold, silver] [silver, silver] 2. [gold, ----] [gold, silver] [silver, silver] 3. [gold, gold] [----, silver] [silver, silver] In 2/3 of the scenarios, you drew from box 1 and the remaining coin is gold. In 1/3 of the scenarios, you drew from box 2 and the remaining coin is silver.
- MarkMc 5y agoTo me the best explanation of the Monty Hall problem is to imagine there are 1000 doors. You pick a door and the host opens 998 other doors. Since you almost certainly did not choose the correct door, the host has almost certainly identify the correct door, so you should change your guess. Then apply the same logic for 100 doors, then 10 doors, then 3 doors.