Monty Hall II: Revenge of Monty Hall

This is a follow-up to my original video on the Monty Hall Problem. Monty has realised we've worked it out and is tired of people always swapping doors and winning so often. So he has an idea...
The original gameshow was called "Let's Make A Deal", hosted by Monty Hall. Monty didn't always offer a switch, and if he did it was for a lesser cash alternative. Monty himself understood 'The Monty Hall Problem'. Here's an interview with the man from 1991 www.nytimes.com...

[@NovaTheReaper]

It's always amazed me how much easier to understand math seems to become after introducing a few charts and some checkmarks and x's

[@ericsurf6]

I've always wondered about this one. Thanks so much for the probability education. You're a genius! Can I subscribe twice?

[@skycanth1969]

If my car was the prize you'd want to win the goat every time

[@singingbanana]

Good question. The show was called "Let's Make A Deal", hosted by Monty Hall. In the gameshow itself Monty would reveal 'a goat', but then wouldn't necessarily offer a switch. And he didn't offer what was behind the other door, but a lesser cash alternative. Monty himself knew of the 'Monty Hall Problem' and understood it. I've put a link in the description of an interview from 1991 with the man.

[@seancunnion1438]

I want to see "Monty Hall II: Revenge of Monty Hall: Attack of the Goats"

[@janinipizzicato]

Would be the best movie title ever XD

[@phjorland]

Followed by "Monty Hall III: Return of the Car"

[@aaronfowler4921]

If you won a car in the past you must return it for a goat in equivalent condition?

[@sebastianespejoloyaga7603]

Peter Hjørland More like Monty Halll, am I right?

[@fadyserhan9537]

U should see the movie "the return of attacking tomatoes"

[@NoriMori1992]

2:15 "And it pretty much sucks to be you." I laughed so hard at that.

[@singingbanana]

Ah thanks, you have just made our AV guy very happy. The camera was a proper one, rather than my usual home camera, I will ask. I believe he edited it in final cut pro, I can probably get more tech details. At home I used cheaper editors.

[@TyYann]

1:40 "You lose and you go home with a goat". I don't call that losing, I call that winning a goat... ;~)

[@MissMisstreater]

love it - this was my argument all along, even in the Monty Hall situation, you can never know if you're host is out to get you! Though saying that, it still probably makes sense to switch then as you either have 50-50 or 76-33, switching never hurts your odds

[@marcochimio]

I think you meant 67-33.

[@singingbanana]

That's a genuine blackboard here at Cambridge maths department :)

[@singingbanana]

Adrian says it was shot with a Sony HVR-Z1E camera, edited with final cut pro but have a look at the new Panasonic P2 camera if you're interested in buying.

[@floyo]

This question is equivalent to Monty opening a random door and asking you to switch or not. If he happens to open the car then you always get a goat. This is similar to you picking a goat and him trowing tails. In this case, by symmetry, it is obvious that it is 50/50

[@stormsand9]

And then... The third problem. Backstage, monty plays rock paper scissors with an employee. If he wins rock paper scissors, THEN he will get to flip the coin. Possibilities are endless

[@jvcmarc]

So, I've done the math: in average the probability of winning by sticking is of: 1/3. and the average probability of winning by switching is of: 1/2. with 1/6 being the instant lose case. that is considering that both the rock paper scissors and the coin flip are completly random (that is, the last outcome doesn't affect the next one. (as it could happen in a normal game of rock paper scissors.))

[@Zororo23]

explain it to me pls

[@Felixr2]

If the last outcome doesn't affect the next one, the chance of winning rock paper scissors in a 1v1 match is still exactly 0.5, the same as throwing heads with a coin flip. So the odds remain exactly as shown in this video.

[@singingbanana]

Aargh! Gremlins!

[@lucas_teo]

You made mathematics look so much FUN!!

[@singingbanana]

I know the one. I'll think about it and try and say something intelligent about it.

[@singingbanana]

Glad you tried it. Some people need to be told to pause if they want to try it. You got the answer and it was wrong?

[@stephenm6165]

What if I want a goat?

[@junkjunk2493]

ok funny guy , show your self out

[@Strijdparel]

That is so cool, I never got the Monty Hall thing, but after wachtching your vid i got it instantly

[@singingbanana]

D'oh! But really glad you had a play with it.

[@singingbanana]

This was made on Adrian's machine. I got the background by spending eight years studying at university then getting a job at Cambridge. Then finally standing in front of the blackboard in the coffee room.

[@rocykel]

Not so! Blackboards are still common at our universities. Whiteboards can't be wiped clean properly (at least not without dedicated solvents) and it's all too easy to use permanent markers instead of whiteboard markers on them. Whiteboard markers are expensive and don't last long, and they dry out quickly if left without a cap. Chalk is pretty dry to begin with! And chalk is cheap, and pure water will get rid of it every time.

[@joeytje50]

I've actually thought of another way to make this 50/50 myself: in the original scenario, you pick a door, and a goat-door gets opened. Someone else then comes on the stage without knowing your first pick. You both say which door you'd like to open then. He will have 50% chance of winning, and you will still have 66% or 33% chance of winning. You're picking from the exact same 2 doors, but you can still have a better chance of winning.

[@romanr9883]

attack of the goats lol

[@KarstenJohansson]

One way I prove the monty hall problem is to use all 52 cards. I secretly write one down and then ask the spec what the chances are that they'll pick that card randomly from the 52. They correctly identify that it is 1/52. Then I get them to choose one and put their finger on it. I remove ALL the cards except 1. My card or theirs is the right card. What are the odds that they chose the right one now? 50/50? Not a chance in hell. They always recognize the odds are highly in their favour if they change to the card I singled out since the odds are still 1/52 that they chose the right card - 51/52 that they were wrong. Funny how using only 3 cards makes this so hard to recognize.

[@lets_see_777]

thats minutephysics or some other video

[@tajshoosh1196]

This is one of the better explanations of the original Monty Hall. The issue missed by those puzzled by the statistical logic is that the game host has “information”: she knows the location of the goats & the car. So the host is not eliminating a “random” option among the 3, but a “known” outcome (goat) from 2 identical outcomes (goat1 & goat2). This makes switching a better option with 2/3 probability, instead of the intuitive 1/3. Some are also confused by the gut feeling that 2/3 probability means 100% certainty. The random process of tossing a coin erases the impact of that information. Therefore, the probability is reduced from 2/3 to 1/3.

[@Martymer81]

They still use blackboards at Cambridge!? I thought all universities used whiteboards nowadays... Hell, in Sweden even elementary schools have made the switch. I'll assume it's not a matter of money, so what's the reason? Tradition? Just curious :)

[@singingbanana]

We like blackboards.

[@xXNuclearWarXx]

@peter8aus8berlin You're supposed to assume that the host always reveals a goat and always offers you the opportunity to switch, not just if you picked the car. Under those circumstances you *will* win 2/3 of the time by switching.

[@kyle7447]

I like this new version! As soon as you said flip a coin i got it haha.

[@lamonthall82]

This is the comedian Monty Hall who can appreciate wisdom😀

[@guyfromkerala3577]

Ok....but in the first case in which we have selected the car. Is there a need to consider getting a head or a take. It is sure that he will offer a switch. Why should we then take two sub cases? Plz clear my doubt.

[@WiseGuy508]

Hey James recommended your channel in a Numberphile video and then I subscribed to you.

[@peter8aus8berlin]

The reader's letter which the Monty Hall Problem is based upon does not specify wheather Monty offers the switching invariably (and if so whether the contestant knows this in advance). Looking for the average of the highest and the lowest chance a contestant can have I came of cause to the very same result: Monty offers only to switch if you pick the car. sticking=1/3 swapping=ZERO coin = 1/6 Monty offers you to switch invariably. sticking = 1/3 swapping=2/3 coin=1/2 Average is always: 1/3

[@Error081688]

Well done!

[@XaxtonRevolution2]

I don't have to go home with a goat, I can just refuse to accept the prize, and it doesn't necessarily suck to be me, because I might want a goat.

[@hornylink]

+Xaxton Revolution don't "go home" with a goat, you'll ruin your bed

[@NoriMori1992]

The underlying assumption of the problem is that all players want a car, and any other outcome is undesirable. Obviously, not everyone in the world is going to feel that way. But we have to assume they do for the purposes of this problem.

[@badmanjones179]

xkcd.com/1282/

[@Radditz770]

Pretty simple when you think about it. There are 12 different ways it can play out (so 12 outcomes). 4 of those is you winning. 4/12 = 1/3 so you got one third's chance to win.

[@XJoschYZ]

Why would I loose when I pick door 1 and the car is behind door 1? Even if he throws tails he will just show me the car and I win. In this Version the only difference is the limited possibility to switch. You should still switch whenever possible.

[@wisemantellsyousomething1134]

0:36 That is absolutely NOT a goat. It's a freaking Mogwai. And it's far more valuable than a car!!! I want one, period.

[@bentoth9555]

+Wise Man (tells you something) I dunno, the rules aren't very clear on those things. I mean don't feed it after midnight, sure. But is that midnight local time? And if you get the Mogwai in, say, California, then travel to London with it, does it still go off local time? Or is jet lag a factor and it goes off midnight Pacific time?

[@wisemantellsyousomething1134]

+Ben Toth What's your point? Just give it to me! :D ...Oh, wait, you're right. When to feed it then...? Well, I guess, its "magic" is triggered from the highest stand of the moon (midnight), until sunrise. Yeah, that makes sense... I'll go with that. :)

[@bentoth9555]

I'm just thinking it should be a bit clearer given the risks involved. :P

[@bentoth9555]

That wasn't a goat behind number 3. That's a mogwai, you're not gonna fool me. :P

[@ArsonBeanTanks]

I'd feel stupid if I switched doors and it was the wrong door that time.

[@PowerStar004]

This is similar to the Monty *HELL* problem but with an extra coin flip to make it more mathematically interesting. The Monty Hell problem is a senerio in which Monty really wants you to lose. If you pick a goat, Monty will ALWAYS open your door, and you lose, period. If you pick the car... Monty will then offer you the chance to switch. Obviously, since he only offers then if you chose right to begin with, you should Never, ever switch. Unsurprisingly, your overall odds of winning Monty Hell are rather low, even if you know not to switch. The reverse, where Monty really wants you to win, "Monty Heaven", involves Monty only offering you the choice to switch if you choose a GOAT originally - if you choose the car, he will reveal the car and declare you the winner on the spot. Yes, your overall odds of winning Money heaven are high, in fact, if you KNOW you're playing Money Heaven you can't lose!

[@snakerattleroll6678]

Wouldn't the Monty Hell Scenario be all three doors having goats, but one is cleverly disguised as a brand new Car?

[@hainsay]

Monty Hell would just be a 1/3 chance of winning if you know what he's doing.

[@blozier2006]

BULL. Monty isn't tired of people switching and winning... he is tired of people taking the scenario out of context, since Monty NEVER allowed you to switch doors, he only offered a cash buyout, basically asking you to FORGET the doors altogether. go to /watch?v=c1BSkquWkDo and hear this from the man himself.

[@MauveTendingToBeige]

It is simpler to keep Monty in the dark. As in your proposition 1/3 of the time the game will end instantly when the uninformed Monty opens the door to the car. And 2/3 of the time a switch can be offered (or not, doesn't matter) with a 50/50 chance. No coin tossing, same result.

[@GetMeThere1]

There's a problem with the entire logic of this that people are not giving consideration to: The value of a fine and sturdy goat is being severely underestimated. Just think, there are places where a car would be worthless, but in which you could trade that goat for a fine and sturdy wife.

[@fishooi]

But I thought that if you picked the right door and Monty flips a tail and you win directly. If that's the case then you don't get to switch so chance of losing in this scenario is zero and you'll be back to same case as before.

[@EliosMoonElios]

The percent can't be added. If a factor change, all the ecuation change too. If the guest show the car you still have the 33%? And 30/70(any side) is the normal result of the 50/50 odds.

[@scotland7yard]

yes, for some reason I applied the original MH problem scenario when the car was not behind door 1 and MH got heads with the coin... and I applied the new scenario for the rest of the cases... what made me think that (no car, heads) was equivalent to the original problem is beyond me, rats :)

[@TyYann]

Thanks!

[@Markinpuff]

All that really matters is the first pick, the odds are against you. Switching is obviously the play.

[@themeeman]

Wait what an HD video uploaded in 2009? This makes no sense.

[@Nintendo101maniac]

Stick with your answer. If it's the car you're going to get the switch 100%, if it's the goat, you'll get the switch 50%. So you get 2 cars and 2 goats. Both cars you get a switch, one goat you get a switch the other you get. So staying, you have a 33% goat and 66%car. Basically, the coin trick made it a little more obvious by unbalancing the probabilities.

[@RonaldABG]

You got it wrong, 1) If it's car (1/3), you get the switch 100% of it, which is 1/3 of the total. 2) If it's a goat (2/3), you get the switch 50% of it, which is (1/2)*(2/3) = 2/6 = 1/3 of the total. So, you get the switch 1/3 + 1/3 = 2/3 of the total cases, that are constituted by 1/3 in which you have the car and 1/3 in which you have a goat. You win the car 50% of those cases staying.

[@whytauisrightandpiiswrong3296]

You're wrong, now just admit it and stop posting that same spam comment on every monty hall video.

[@scotland7yard]

nice! i paused at 1:50 to give it a try myself and i got the answer!! ... but it was wrong :(

[@TheKartKiller]

It is now a genuinely random game because there is a coin toss (which is a random experiment) . In the previous one, because there was a human factor involved, the odds favored someone who switched. Am i right to reason it out this way?

[@hainsay]

It's not far off, but also the probabilities were chosen to specifically make it a 50% chance. If Monty had rolled a die and offered a door change if he rolled a 5 or 6 then you are reducing the probability that it is a good idea to switch and it is better to stay.

[@livedandletdie]

hmm I still think there should be yet another Monty Hall which is even more difficult to win. Let's say 4 doors. 1 Car 3 Goats. Door 1: Car, Stick. If Switch 2 lose. Door 1: Car, Stick. If Switch 3 lose. Door 1: Car, Stick, If Switch 4 lose. Door 1: Goat, Stick Lose. If Switch 2 Win. Door 1: Goat, Stick Lose. If Switch 3 Lose. Door 1: Goat, Stick Lose. I Switch 4 Lose. Door 1: Goat, Stick Lose. If Switch 2 Lose. Door 1: Goat, Stick Lose. If Switch 3 Win. Door 1: Goat, Stick Lose. I Switch 4 Lose. Door 1: Goat, Stick Lose. If Switch 2 Lose. Door 1: Goat, Stick Lose. If Switch 3 Lose. Door 1: Goat, Stick Lose. I Switch 4 Win. And now I consider the game fair. 3/12 times stick. 3/12 times Switch. Which means you are Ever so likely to lose. I mean 3/4 chance of losing doesn't sound too bad.

[@mikejonsson7235]

even if the chance double you never know witch door has a car or a goat behind, so it dosent matter witch number you start or end with

[@yousefabdelsamie5983]

Which Uni did u graduate from

[@jakepollen6839]

So this Makes the Monty Python game legitimate since it is truly random?

[@KasabianFan44]

Hey singingbanana, I tried to make up a variation of this problem which looked like this: As before, you have to pick one of the doors. Behind one of the doors is a car and the other two contain goats. Once you've picked a door, the host opens one of the other doors (which has a goat) and lets you choose between sticking with your door or switching to the other one. But now, after you've made your choice, the host is going to toss a coin; if he throws heads, he will do what you wanted him to do (i.e. stick if you wanted to stick, switch if you wanted to switch). While if he throws tails, he will do the opposite (i.e. switch if you wanted to stick, stick if you wanted to switch). The question is: what are the probabilities of winning a car now? Is it still 2/3 if you switch and 1/3 if you stick? Is it now a 50/50 chance? Or is it maybe a 1/3 chance if you switch and 2/3 if you stick? (intuition tells me it's 50/50 but I'm not sure)

[@Whoeverheis11]

Alright, let's make the chart, assume you pick #1, and X marks the car: 1 2 3 Coin Stick Switch X H Lose Win X T Win Lose X H Win Lose X T Lose Win X H Win Lose X T Lose Win Your modified Monty Hall brings your odds to 50/50.

[@KasabianFan44]

Whoeverheis11 Okay, I understand now, thanks for the explanation :)

[@donfrankmagic6442]

Ok After doing a sample space, I believe it is correct.

[@watchdogs1328]

Always account for variable change.

[@Explodenator]

poor little gremlin... he demoted from one vid to the next

[@oO_ox_O]

you might say it's misnamed but apart from that, so what?

[@multicolor0]

or you could just pick a door and have him open THAT door

[@SomeRandomFellow]

Or, a better variation: Three doors. Pick a door. Thats it

[@onlynamelefthere]

you should defintivly clean your blackboard more often

[@arrow1archer]

what pc do you have and how did you get the background

[@ManolitoMystiq]

If I'm not happy, what then?

[@charlieholloway627]

I still want the goat

[@leppy1563]

You know your on the internet when he needs to say Which you don't want to win :L

[@leon123456789ize]

DOES A CAR GIVE YOU MILK? NO,thats what i thought so

[@denmartorlanda]

have you ever seen a singing banana? no, not the channel...

[@hatujemeletsplayeryheskyce6460]

or just pick a door and have 33℅ chance instead

[@johnnyvaughan5841]

I don't think I'd call going home with a goat losing

[@LukeVidalis]

great video!!!!!!!!5 starts

[@theaxehandle1]

how would this work as a real game? Because if Monty Hall only flips a coin when the contestant picked a goat, the contestant would know to stick with his choice when Monty doesn't flip a coin, or to switch whenever he does flip a Heads. And if he always opens your door whenever he flips Tails no matter what you choose, you now have a 4/9 chance to win instead of 1/3 like before.

[@1TieDye1]

He walks off stage, out of sight.

[@mander40101]

There is a glaring problem with any explanation of the Monty Hall problem. On the big deal of the day there are two players, both of whom chose a door. If neither pick the big deal, Monty opens one of the contestants door, usually the one that's a zonk. I watched some videos, and after opening the first door, the contestant is not given a chance to switch. If they picked the big deal, then the door no one chose is opened first, and the big deal is revealed last. If the contestant didn't pick the big deal, the contestants door is revealed, followed by the big deal no one chose. The Monty Hall problem is based on one player, not two, and the contestants on the show aren't given the option to switch doors. I have yet to find a video of Let's Make a deal that's a true reflection of the actual problem as stated.

[@Hank254]

That's correct, the game was never played this way on Let's Make a Deal, it was only named after the game show host. You could call it the Bob Barker problem if you prefer... still a very good puzzle.

[@ghuegel]

Neat idea for a modified Monty Hall problem. A side-puzzle I tried was... in this coin-flip scenario, if I could see the result of the flip, I could boost my odds of winning to well above 1/3. I figure that by switching for heads and not switching for tails, my overall odds of winning would be 50/50.

[@zxjuifothrui]

Then Monty Hall gets a double tailed coin.

[@scotland7yard]

thanks mate. btw, I'd be very interested in hearing your views on the unexpected hanging paradox, one of the few problems in which the more you think the more you wanna go "aaaahhhhggg!!"

[@rossmccluskie]

Many thanks man. This information is very helpful. (Say thanks to Adrain too!) Really love the singingbanana channel.... look forward to every new video! Take it easy!

[@rossmccluskie]

Hey singingbanana... the quality of this video is excellent. I'd love to know what device you used to film this and what you use to edit and export your videos??

[@Bratzbeerkuchen]

Would it be same with more doors?

[@SpectatorAlius]

In game theory, what Monty does in this version is called a "mixed strategy". The previous version of the Monty Hall problem, the one Numberphile did in their other videos, has both Monty and the player following a "pure strategy". What is a little surprising is that once Monty chooses the mixed strategy, it no longer seems to matter which strategy the contestant chooses.

[@avinandanbanerjee9568]

Can we alternatively use Baye's theorem?

[@singingbanana]

Yes, but it's a little messier than typical Bayes Theorem en.wikipedia.org/wiki/Monty_Hall_problem#Direct_calculation

[@aaronfowler4921]

I don't think he needs to go back stage or even know what doors have what prizes at this point though. You just add the rule if he opens the door with the car you lose. So the game would be you pick a door, he randomly chooses a different door if there is a car you lose if there is a goat you get the option to change your answer. This is the same probability right?

[@RonaldABG]

If the host does it randomly, once a goat is revealed the chances are not 2/3 for switching, and the reason is we are in a subset of the cases, and in a subset the proportions are not necessarily the same. Supposing you play 99 times, on average you choose the car in 33 and a goat in 66 of them. Then the host opens a door, but note he would reveal the car in half of the cases you chose a goat (in 33). So once a goat is revealed you know you are not in those 33; you could be in the 33 you chose a car or in the other 33 you chose a goat. Therefore you win half of these cases staying and half switching. As an analogy, since the host does not know, it is the same if who decided which door is going to be revealed was the contestant, because he is doing it without information. So suppose the contestant chooses the two doors that are going to remain closed and then the third one is revealed and it results to have a goat. Which of the two doors have more probability and why?

[@mustacheglasses5765]

The illustration was helpful. I heard of the Monty Hall Problem and the solution. Before I wrote out all the possibilities I couldn't believe that switching would net me a better chance of winning the car.

[@dushas9871]

you do understand that he's talking about different problem here, right? In the original problem you do have better chance of winning if you switch.

[@mustacheglasses5765]

...yes. The problem has changed. I was simply saying that when I heard others talk about The Monty Hall Problem they didn't explain it in a way that made sense to me. Whereas he has explained this problem (which is slightly different) in a way that was very clear and beneficial.

[@vikasmishra1190]

Mustache Glasses gggghgbbbbvgbbnnnbbhhbhhhh

[@IronMaytons]

Thats not a goat ... Thats Gizmo!!!!!

[@JakobVirgil]

I have been saying this for years. I think it was partially the root of the misunderstanding of Marilyn's presentation of it.

[@armacham]

0:42 If the original game was genuinely random, you would still have a 2/3 chance to win by switching. Genuinely random meaning the car is equally likely to be behind doors 1, 2, and 3, and the host is obligated to reveal a goat, and when the player chose the correct door to begin with, the host randomly chooses between the other two doors to decide which one to open.

[@yurenchu]

This variant is essentially the same as the Deal Or No Deal scenario (or, in other words, the scenario in which Monty would, with a 50% likelihood, reveal the car if your first pick was a door with a goat).

[@grendelum]

The most uses of "you're going home with a goat" in a KZbin video... were you actually able to claim the goat?

[@Nulono]

3:30 What's that in the middle?

[@donfrankmagic6442]

I believe this explanation is incorrect.

[@PolarBearMath]

I LOVE YOU BANANAMAN! :D

[@HeyKevinYT]

genius

[@PolarBearMath]

P.s. your goat ish a gremlim D:

[@rymdmyggan]

why is it always goats and a car? why cant it be piles of shit and a bar of gold or something? does the monty hall problem only work with goats and cars and if its on a gameshow?

[@mrkappa7995]

My Brain hurts 😵🔫