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  • Hello Ding-a-lings, my name is Michael and today we're going to discuss the Monty Hall

  • Problem.

  • But why?

  • Why should anyone discuss the Monty Hall Problem.

  • It has been discussed ad nauseam.

  • There are so many articles and videos about it that are fantastic including Kevin's

  • recent video about paradoxes on Vsauce 2.

  • Check that one out, please.

  • Does the world really need another discussion of the Monty Hall Problem?

  • No.

  • But maybe it does.

  • You see I believe that true understanding comes from listening to as many voices as

  • possible, discuss something from as many perspectives as possible and so the more people you listen

  • to talk about the Monty Hall Problem and why it's solution is what it is, I believe through

  • osmosis the greater your understanding will be and lately for the last few weeks I have

  • been driving around on my commute to and from the Vsauce office talking out loud to myself

  • ab out the Monty Hall Problem.

  • If you see me talking to myself in my car I'm not on the phone.

  • I'm literally explaining the Monty Hall Problem to myself, honing in on the best way

  • to intuitively get it and I feel like I have found an explanation thatwell that I like.

  • I'm not saying it's the best but I wanted to share it with you in the spirit of understanding.

  • So let's dive in.

  • First of all what is the Monty Hall problem.

  • Well it was named after game show host, Monty Hall who hosted let's make a deal.

  • One of the challenges he would put the audience to involved three doors on the stage: door

  • number one, door number 2, and door number 3.

  • Now Monty would honestly tell the audience member that behind two of the three doors

  • was a goat.

  • A goat.

  • Yeah.

  • But behind one of the doors was a bunch of money.

  • Let's say a million dollars.

  • Now the audience member got to pick one of the three doors and if they picked the door

  • that had money behind it they would get to keep the money but if they picked a door that

  • had a goat behind it, well they'd get a goat and they'd have to like take care of

  • it and travel back home with it.

  • Point is, you weren't supposed to want the goat but goats are awesome she I, in my version,

  • have replaced the goats with pieces of poop.

  • Fewer people want poop.

  • Some still will but the point is you're supposed to want the money.

  • If you pick a door that has poop behind it you have to, like the goat, take the poop

  • home, and take care of it and feed it, all that normal poop stuff.

  • So here we go.

  • Uhh door one.

  • Door one.

  • Okay.

  • Awesome.

  • Before I open door one, I'm gonna tell you this.

  • It's good that you picked door number one because there's poop behind door number

  • two.

  • You see that?

  • There's some poop.

  • Oh you don't want that.

  • But now, Hannah before I give you what's behind the door that you have chosen I'm

  • gonna actually let you switch.

  • You can switch to door number three or you can stick with your original choice.

  • After you've done that, either you've switched or not, you will have to take home

  • whatever is behind the door that you have chosen.

  • Would you like to switch or stick with your original choice?

  • I'll stick with my choice.

  • Why?

  • Because it doesn't matter.

  • It doesn't matter.

  • It doesn't matter right?

  • It kinda feels like a 50/50 right.

  • You know that there's no money behind door number two which means the money's either

  • behind door number one or door number three.

  • So you have a 50/50 chance right?

  • I mean why not just stick with your first choice because what if your first choice was

  • right and ya switched.

  • Well then you would really hate yourself for not going with your gut.

  • It would feel like you lost the money.

  • Like you had it and then you lost it by switching.

  • If it's behind the door you didn't pick then it's like you just picked wrong.

  • You don't feel like you're losing something so psychologically yes most people decided

  • not to switch.

  • But mathematically you should always switch.

  • In fact if you switch you will win not half the time, it's not actually 50/50.

  • If you switch you will win 2/3 of the time.

  • And if you don't switch you'll only win 1/3 of the time.

  • So you have a 66.6 repeating chance of winning if you switch.

  • Why should that be?

  • Why is it not 50/50?

  • Well it comes down to two extremely important facts about this problem.

  • The first one is quite easy for us all to agree on.

  • It's the fact that when you make your first choice, oh by the way I never actually showed

  • you the result.

  • You stuck with 1, and Hannah just like I said, you should have switched.

  • You shouldn't always switch, let's be clear about that.

  • I'm not telling you a strategy to always win.

  • I'm just telling you the best strategy in the long term.

  • So you chose to stick with one but there's poop behind one.

  • The money was behind three.

  • So there it is, a million dollars to scale.

  • So, let's get back to these two important facts.

  • The first is that when you first choose a door the probability that you have chosen

  • the money is 1/3.

  • Okay that's the chance that you'll be right.

  • The money is behind one of three doors.

  • You have a one in three chance of picking the door with money behind it right off the

  • bat.

  • That's pretty clear but that also means 2/3 of the time you are choosing a door that

  • does not have money behind it.

  • More doors have poop behind them than have money behind them.

  • The second and much more important fact is that the host knows which door the money is

  • behind and the host will never open a door that has money behind it.

  • I believe most disagreements over the solution of the Monty Hall Problem come from confusion

  • over the rules of the game.

  • So let's make it extremely clear.

  • First you pick a door.

  • Next the host opens one of the two doors you did not pick.

  • And the door the host opens of those two will always be a door with the goat behind it.

  • You are then asked if you would like to stick with your original choice or switch to the

  • door that you didn't pick that the host also didn't open.

  • This is why it feels like a 50/50.

  • We know that the money is either behind the door you picked or the door that the host

  • didn't open that you didn't pick.

  • It's behind one of them because it's definitely not behind the one the host opened.

  • The host only ever opens a door with a goat behind it.

  • But surprisingly to switch or not is not a 50/50 choice.

  • Switching allows you to win 2 out of 3 times.

  • If you chose a door like Hannah did, door number one say, and then the host flipped

  • a coin and randomly opened one, sometimes the host would open a door that had the money

  • behind it and that would be terrible television because the money would be revealed and the

  • host would say okay well you were wrong, would you like to switch to the one door that remains

  • closed.

  • Of course you wouldn't.

  • It wouldn't matter.

  • The game would be over and it would be over.

  • Instead the host always opens a door of the two you haven't chosen that has poop behind

  • it.

  • Okay.

  • Or a goat depending on which version you wanna go with.

  • So what does that mean?

  • Well it means that sometimes you choose the door that the money is behind.

  • That's awesome.

  • And in that case it doesn't matter which door the host opens.

  • The host could have a rule where they always open the left most door or maybe they just

  • flip a coin and open one or two.

  • Either way, the host can open either door.

  • But most of the time you didn't pick the door the money is behind.

  • 2/3 of the time you pick incorrectly.

  • In that case the host is forced to open one door.

  • The host has no choice.

  • The host cannot open three because the money is behind that door.

  • The host must open the other door to reveal the poop.

  • This is why you actually have more information than you might think.

  • The host is always avoiding the door with the money behind it.

  • So by looking at which door when you have the choice to switch is still closed.

  • You actually know a little bit more about what might be behind that door.

  • In the case that you picked correctly the first time, you picked a door with the money,

  • the host has decided to open one door and not the other simply because of random chance.

  • But most of the time you're choosing incorrectly and the host has left a door closed.

  • The door that the host left closed was left closed because there's money behind it and

  • you should switch.

  • Okay.

  • To make this really really clear, if I choose correctly, the door that remains closed has

  • no money behind it.

  • If I don't choose correctly the door that remains closed does.

  • So if I'm wrong with my first choice I should switch.

  • If I'm correct with my first choice I shouldn't switch.

  • Okay?

  • Perfect.

  • Here's the thing.

  • You choose correctly much less often if we were to run many many trials you choose correctly

  • much less often than you choose incorrectly because you're only picking the door with

  • the money behind it 1/3 of the time.

  • 2/3 of the time, most of the time, you will have chosen a door with poop or goat, maybe

  • it's goat poop behind it.

  • So most of the time you should switch because the door that remains closed is the money

  • door.

  • There's an analogy that I think makes this even more clear.

  • And it involves a sack.

  • This sack contains 3 marbles.

  • Two are white and one is black.

  • We can say, let me put them on the bag, great.

  • Okay now we can say that the goats are represented by the white marbles and the million dollars

  • are the black marble.

  • You want the black marble and when things begin I don't know which marble is which.

  • I just have to reach in there and pull one out and after I've pulled on out I don't

  • get to look.

  • Perfect.

  • I've pulled one out.

  • I don't know if this is a white or black marble but I want the black one.

  • Remember that.

  • But now what happens, an analogy to the Monty Hall problem with money and goats is that

  • I give this bag to the host.

  • The host then looks inside the bag and pulls out a white marble.

  • The host will always pull out a white marble because the host will never reveal the black

  • marble.

  • The host will never open the door that has the money behind it and there will always

  • be at least one white marble left in the sack because if I chose the black marble then there

  • are two white marbles in here.

  • If I chose the white marble then there's a white and a black.

  • Either way the host has a white marble to remove.

  • The question now is should I switch?

  • Should I go with the marble I initially picked?

  • I don't know what it is.

  • Or should I switch tot eh bag?

  • I should always switch to the bag because I'm much more likely to have chosen a white

  • marble.

  • The host then removes the other white and there's a black marble in here.

  • If I choose correctly with my draw that means I've pulled out the black marble and there

  • are two white marbles in here.

  • Well then of course when the host removes one white marble there's still a white in

  • the bag and I shouldn't switch but I choose the black marble initially only 1/3 of the

  • time.

  • Most of the time I am handing the host a bag that contains the black marble and one white

  • marble.

  • The host's job then is to remove a white marble, leaving most of the time the black

  • marble that I want in the bag and so I should switch to what's in the bag and not what's

  • in my hand.

  • Now you should be prepared to make a killing on let's make a deal.

  • Very good.

  • Stay curious, keep exploring probability and as always thanks for watching.

  • Oh yeah!

  • Oh I'm grr baby grr.

  • Get it?

  • It's Borat.

Hello Ding-a-lings, my name is Michael and today we're going to discuss the Monty Hall

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A2 初級

モンティホール問題 (The Monty Hall Problem)

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    林宜悉 に公開 2021 年 01 月 14 日
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