The Monty Hall problem

Retro

Founder
Staff Member
Joined
4 Jun 2021
Messages
6,153 (4.46/day)
Location
UK
Here's a great explanation of the Monty Hall problem with great graphics and clear dialog on the Fexl channel.

To view this content we will need your consent to set third party cookies.
For more detailed information, see our cookies page.

If you want to read about it:

 

Mars

Moderator
Staff Member
Joined
10 Jul 2021
Messages
622 (0.46/day)
Don't have time to listen to it all now, but I have immediately found a flaw in the explanation: in the 100 doors example the 'skipped' door has a 99% chance of having the car behind it, while the gamer's original choice, door 1, still has its piddly 1% chance. Really?

I don't agree, and here is why: with all the doors having been eliminated, and only TWO doors left, the chance of the car being behind either of them is 50/50. NOT 99% vs 1%.

I shall definitely come to it later and watch it to the end, see what I think then.
 

Retro

Founder
Staff Member
Joined
4 Jun 2021
Messages
6,153 (4.46/day)
Location
UK
Yeah, I agree, something doesn't compute there and it confused me. When that happens, it's usually because something doesn't make sense. Didn't want to say anything in the first post to avoid biasing readers. However, this is somehow proven mathematically so the Wikipedia article may help. I still wanna fully get my head around it before I accept it.

I think Veritasium also did a video on this, so I'll try to find it and post it here.
 

Hitcore

Well-known member
Joined
11 Feb 2025
Messages
49 (1.53/day)
Location
Legoland
Here's a great explanation of the Monty Hall problem with great graphics and clear dialog on the Fexl channel.

To view this content we will need your consent to set third party cookies.
For more detailed information, see our cookies page.

If you want to read about it:


With two doors left, there is a 50/50 shot, no matter how much they overcomplicate this. I'm the kind of person who can't be bothered with some TV show host's cheap mind tricks and I would just stick with the initially chosen door, I am stubborn like that. If I win a goat, I actually take the goat back home with me, I have a big garden and barn, he can live there. I will name the goat Dennis and I will love him. If I win the car, it will be of no use for me as I do not own a driver's license, and I will most likely end up selling the car, and will buy a goat, possibly the same goat from the show even, because I really like Dennis. Ergo: I always win. Mathematicians hate me.
 
Last edited:

Retro

Founder
Staff Member
Joined
4 Jun 2021
Messages
6,153 (4.46/day)
Location
UK
Mathematicians hate me.
Haha I love that. 🤣

Thing is, it's been mathematically proven, allegedly. Normally when things like this don't quite make sense, I let it slide after thinking about it a bit. Gonna make an exception in this case though and make the extra effort to understand all the angles and come to a more informed conclusion. It still looks 50/50 to me too.
 

Hitcore

Well-known member
Joined
11 Feb 2025
Messages
49 (1.53/day)
Location
Legoland
So picrelated is what's causing them to say: "ThErE aRe MoRe WiNNinG sCeNaRiOs WiTh sWiTcHiNg So MaThEmAtTiCaLLy tHiS iS pRoOf!!!!11one"

Screenshot_20250308-231926.webp


But if we look closely they are lumping together staying winning reveals into one (Door [x] or Door [x]). It may sound trivial but technically they are different scenarios. Individualize them on the silly chart and we get the same amount of wins as losses.
50/50.
I just broke the internet.
 

Retro

Founder
Staff Member
Joined
4 Jun 2021
Messages
6,153 (4.46/day)
Location
UK
Ok, I've now looked at it properly, critically, I've also drunk the koolaid, and can confirm that the theory is right: you should switch every time to improve your odds to 2/3.

Without going through the whole explanation again, the core of it is that when you pick the initial door, you know nothing about what's behind any of the doors, making it 1/3 that you'll win the car. When Monty opens the door is when critical new information comes in that changes the odds: he will always open the door with the goat behind it since he knows where the car is. If he didn't know where the car is, then the probability would become 1/2 like people usually think initially as sometimes he would open the door with the car behind it, but that never happens. Therefore, when Monty opens the door, the 2/3 probability is concentrated in the unopened door. In the 100 door version, it becomes easier to see, since the 99% probability is now concentrated in that one remaining unopened door. Once you understand this, the whole thing falls into place.

Here's a handy online simulator to demonstrate the game in action, proving the theory. I like the way it allows you to play with the various options, including simulation speed. I suggest setting the runs to 1000 and speed to instant for the clearest results.


Here's 4 Numberphile videos explaining it in various ways, which helped me to understand how this works.

In the first video, a nice lady, Lisa Goldberg, an adjunct professor in the Department of Statistics at University of California, Berkeley, explains it:

To view this content we will need your consent to set third party cookies.
For more detailed information, see our cookies page.


In the second video, she explains it with some confusing squiggly math. It's basically the same video with the maths bits left in:

To view this content we will need your consent to set third party cookies.
For more detailed information, see our cookies page.


In the third video, Brady Haran, the maker of these Numberphile videos, explains it in just 4 minutes in a clear, intuitive way:

To view this content we will need your consent to set third party cookies.
For more detailed information, see our cookies page.


In the fourth video, Brady explains it in just 48 seconds! It's so efficient, love this.

To view this content we will need your consent to set third party cookies.
For more detailed information, see our cookies page.

Finally, looking at these other videos and the simulator, I think the original video's explanation in my first post is too long at almost 20 minutes and I did notice that he repeated himself several times, which only causes confusion. It actually only takes 5 minutes to explain, a bit more with the squiggly math thrown in as a bonus.
 

EGPRC

New member
Joined
9 Mar 2025
Messages
1 (0.17/day)
What creates the disparity is the fact that the host is not revealing a door randomly but instead it is assumed as a rule that he knows the locations and purposely avoids to open both the player's chosen door and which contains the prize. He always removes doors until two are left, taking care of not removing any of those two.

That means that whenever the player starts failing, the host is inevitably indicating where the car is: in the only other option that he avoids to remove from the rest. With 100 doors, the player would pick wrong in 99 out of 100 attempts, on average, and as the prize is not allowed to be revealed anyway, in those 99 out of 100 times it will be in the other door that the host purposely left closed.

It's like if a second player came to play, he cheated by looking inside all the doors that the first did not pick, and took which preferred from them. In that way, it is obvious that the cheater will pick the winner as long as it is any of the 99 non-chosen ones, so 99% of the time.
 

Hitcore

Well-known member
Joined
11 Feb 2025
Messages
49 (1.53/day)
Location
Legoland
I've watched every video, Retro, and I must say: that is really interesting. But... theory is nice 'n all, so I wanted to test it in practice. I went to the simulator:


20250309_025334.webp

It seems that no matter what I do, whether I decide to change or keep, 9 out of 10 times I get the goat.
To me it is clear now that the universe wants me to fulfill my one and true purpose in life: to become a goat herder. 🐐🙏

24f99996-d540-4a17-b4fc-dbae0bea4917.webp
 
Last edited:
Back
Top Bottom