The Monty Hall problem

[Mars]

I do not accept their explanation.
As I said, yes, I did look at the explanation, and I do not agree that after Monty opens his goaty door the probability suddenly becomes 2/3.
Why 2/3? That lady mathematician just stated it as a fact, 'shrinkage of probabilities', probability concentrates in door 3.....something like that; not good enough.
You seem to understand it, can you explain it so that we understand?

 

[Retro]

I do not accept their explanation.

So, you continue to not accept a rigorous mathematical proof, the one thing you and no one else, including me, can deny? You're actually claiming it's wrong. Really? I'm therefore not gonna go down an endless rabbit hole with you as you won't accept anything I say, wasting my time. In addition, I've already explained it in post 7*, so you already have your explanation from me, but you've just told me that you don't accept it, so, again, you clearly won't accept anything further and I'd be wasting my time, so I won't bother.

*Everyone: it's all there in post 7 with some great explanations from various mathematicians (maths professors no less), an online simulation showing this 2/3 probability in action, plus including a bit from me to help people understand it a bit more easily.

 

[Geffers]

@Geffers It really is 2/3 not 50:50 as it's been mathematically proven. I know it's confusing to get one's head around it, it took me long enough too and I also thought 50:50 initially, but it's worth the effort. I made the post below with a comprehensive explanation of how this works which will clarify it. There's quite a lot to read there, so I recommend fortifying yourself with a cup of tea first.

Some people found the last video, just 48 seconds long, was enough to get it, so perhaps start with that.

The Monty Hall problem

Here's a great explanation of the Monty Hall problem with great graphics and clear dialog on the Fexl channel. If you want to read about it: https://en.wikipedia.org/wiki/Monty_Hall_problem

nerdzone.uk

Also have a look at this post afterwards:

The Monty Hall problem

Here's a great explanation of the Monty Hall problem with great graphics and clear dialog on the Fexl channel. If you want to read about it: https://en.wikipedia.org/wiki/Monty_Hall_problem

nerdzone.uk

Is my reasoning wrong? If so how? Complex answers sometimes confuse what is a simple issue. If you don't know Monty will remove one of the doors I understand the complexity but once that is known only two doors will be ultimately be options and one contains the car and one contains the goat. Still appears to be 50/50 to me and is a wee bit like that algebra formula that seems to prove 1 equals 2 (link below).

Algebra was never a strong point of mine but of course the complexity and the equation seems quite convincing.

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Geffers

 

[Retro]

@Geffers look, it's all comprehensively explained in post 7 that I linked to. Have you properly looked at everything on there? It doesn't look like it.

That there are paradoxes in maths in no way invalidates this, however. Did you know that there's a big, fat inconvenient paradox right at the heart of set theory?!

 

[Geffers]

Watched the video and shall ponder more. It is quite amusing that the complicated answer is considered correct by some whereas others believe the simple answer is the one to go for.

In my younger years I worked for 12 years in betting offices, also used to frequent casinos so am well familiar with odds and systems to supposedly improve 'the odds'.

 

[Retro]

If you see my second post that I linked to, you'll see that I explained it's just a long winded way of opening two out of three doors to make good TV, and it worked before, too. With both ways the chance of winning are 2/3, as long as the player switches.

Oh yeah, betting offices and casinos are the biggest con ever.

If you watch everything and it's still not quite clear to you, then ask here and myself and the members will try to clarify it.

 

[Hitcore]

In the previous page I was sceptical. Being sceptical is my standard modus operandi. And I simply brushed it off with some goat jokes. However, I'm not rigid, and I have slowly come to accept the 2/3 explanation. Let me try to unwrap this in the dumbest way possible, to see if got this right:

Me: I pick door #1.
1 in 3 chance it's the car, 2 in 3 chance it's a goat.
That means: I probably picked a goat.
In comes Monty, he knows where the car is.
He opens, let's say, door #2. It's (always) a goat.
Monty says: wanna switch to door #3?
Why yes, Monty, I will switch.
Because Monty doesn't randomly open a door.
He deliberately opens the door with a goat.
My initial choice (door #1) is still a 1 in 3 chance.
Door #3 now has a 2 in 3 chance.
Thanks Monty, you creepy ass Playmobil figure hairdo wearing showboy.
Not a guarantee, but it's the better choice.

 

[Mars]

@Hitcore. I like your little demonstration, and I would like to think that it may also clarify my suggestion that the choice is 1 of 2, 50/50.

So here goes (your words are in Italics): I am with you until this line: "My initial choice (door #1) is still a 1 in 3 chance". This is where the glitch occurs: the odds for your chosen door where initially 1 in 3, because all three doors were closed, so you had no way of knowing what is behind any of them.

Then this "creepy ass Playmobil figure hairdo wearing showboy" comes along and reveals a goat behind door 2.
This changes the odds!!!
Your initial choice (door#1) no longer has a 1 in 3 chance, but a 1 in 2. There are no 3 doors any more, only two.

Your odds have improved, because one door had yielded its secret and is therefore not an 'unknown' any longer. This has now reduced your odds to 1 in 2.

That's why I find the notion of 2/3 incorrect.

 

[Retro]

@Mars Ignores my reply and continues to say the 50:50 nonsense.

That's why I find the notion of 2/3 incorrect.

In your world that might be a notion, but it's been mathematically proved, as in a rigorous mathematical proof, as I've explained over and over, so not a notion in the real world, no. You must therefore approach this from the point of view that you're missing something rather than trying to disprove the 2/3 probability.

It would be so much more helpful if you just look at those resources that I posted, try to understand the problem and then perhaps come back with an intelligent question if necessary rather than doggedly insisting on 50:50. Even if you said that you accept the proof but just don't get it, that would be fine too and no one's gonna take you to task for it. It's confusing and unintuitive as hell as I've said several times now, but don't try to disprove it. That's just embarrassing and why I'm on your back about this. I said it was an endless rabbit hole with you, didn't I?

 

[Hitcore]

Alright @Mars , I'll break down my findings, line by line, maybe you'll agree then (maybe not).

Me: I pick door #1.
1 in 3 chance it's the car, 2 in 3 chance it's a goat.

At this point the car is behind #1 with 1/3 probability, and it's behind one of the two other doors with 2/3 probability.

That means: I probably picked a goat.

Most of the time (2/3) the initial pick is wrong - it's a goat.

In comes Monty, he knows where the car is.
He opens, let's say, door #2. It's (always) a goat.

Crucial detail. Monty never opens the door with the car. He knows where it is and avoids it.

Monty says: wanna switch to door #3?
Why yes, Monty, I will switch.
Because Monty doesn't randomly open a door.

Monty's action is loaded with information. He is removing one of the two goats from the 2/3 chance pool.

My initial choice (door #1) is still a 1 in 3 chance.

This is still the case. Nothing has changed about my original odds.

Door #3 now has a 2 in 3 chance.

Because the 2/3 chance that the car was not behind my door has now collapsed onto the only remaining unopened door Monty didn't touch.

Thanks Monty, you creepy ass Playmobil figure hairdo wearing showboy.

Fact.

Not a guarantee, but it's the better choice.

Switching gives me a 2 in 3 chance of winning the car, compared to 1 in 3 if I stay.

Now, Mars , I am going to break down your reasoning.

I am with you until this line: "My initial choice (door #1) is still a 1 in 3 chance".

So far so good. You identify the crux of the disagreement: whether the original choice still has a 1/3 chance after Monty opens a door. (it does)

This is where the glitch occurs: the odds for your chosen door where initially 1 in 3, because all three doors were closed, so you had no way of knowing what is behind any of them.

You assume that uncertainty alone determines the odds. That's not how probability works. The 1 in 3 probability isn't just about lack of knowledge: it's about the setup:

• 1 car hidden at random behind 3 doors
• I randomly choose door #1
• That door has a 1/3 chance of hiding the car, regardless of how much I know

It doesn't matter that all the doors are closed. The probability distribution is set at the start.

Then this "creepy ass Playmobil figure hairdo wearing showboy" comes along and reveals a goat behind door 2.
This changes the odds!!!

Revealing a goat does not change the probability that my original choice was right (1/3). It does change the probability distribution across the remaining unopened doors, but not by making it 50/50.

Crucial point: Monty's choice is not random. He always reveals a goat. That means his action is conditional on my initial choice. This preserves the 1/3 vs 2/3 imbalance.

Your initial choice (door#1) no longer has a 1 in 3 chance, but a 1 in 2. There are no 3 doors any more, only two.

Major mistake. Yes, there are only 2 physical doors left unopened, but that doesn't make the probabilities 50/50 (which I believed to be the case as well, previously). What matters is how we got to this point: I picked 1 of 3 doors, 1/3 chance of the car. Monty opened a door that he knew had a goat. That intentional choice funnels the remaining 2/3 probability onto the door he didn't touch (as in: the one I didn't pick and didn't open). If Monty chose randomly 50/50 would be reasonable. But he didn't.

Your odds have improved, because one door had yielded its secret and is therefore not an 'unknown' any longer.

Again, this misapplies intuition. The fact that one door is revealed doesn't redistribute the original probabilities equally. Monty's action is information-rich, not a coin toss. Revealing a goat behind a non-chosen door is more likely if my original pick was wrong. That's why switching is better.

This has now reduced your odds to 1 in 2. That's why I find the notion of 2/3 incorrect.

Nope. If that were true, switching wouldn't matter. But simulation and probability math show switching wins 2/3 of the time.

Mars, I know that your reasoning feels intuitive, as that reasoning was also mine at first, but I've come to learn that we must not confuse fewer doors with equal odds, and we also must not ignore that Monty's action is conditional, not random, or assume that probabilities "reset" after revealing one goat. They don't.

 

[Retro]

Thanks Monty, you creepy ass Playmobil figure hairdo wearing showboy.

Fact.

Totally!

Great explanations, both of them and you gave me a laugh, too. If you can get through to Mars, I'll give you a medal! I'll also give them a gold star, like this one:

Think, if they just let the contestant open two doors without all the faffing, it would be the same 2/3 probability, hence they should always switch, but how boring would that be, so they introduce this confusing element of "choice" on whether to switch or not. Whoever invented this infernal gameshow was an evil genius! It's had people pondering and arguing over it for decades, a masterstroke.