Stumbling Intuition #1: The Monty Hall Problem

[Debater Slayer]

Starting with this thread, I'm planning to initiate a series of threads about counterintuitive problems in mathematics (or programming, if I encounter suitable ones), both for fun and as an opportunity to engage my passion for math.

The original Monty Hall problem is worded thus:

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

Monty Hall problem - Wikipedia

I posted the Wikipedia article to properly cite my source for the above wording, but since it contains the answer to the problem, I recommend not clicking the link before you have come up with your answer!

Before posting your answer, take your time to analyze it! This problem is an example of a veridical paradox, and it is so counterintuitive that Paul Erdős, one of the most prolific mathematicians of the 20th century, wasn't convinced of the correct solution until he saw computer simulations demonstrating its correctness.

A veridical paradox produces a result that appears absurd, but is demonstrated to be true nonetheless.

Paradox - Wikipedia

Also, please only post your answer in spoiler tags so that you don't potentially reveal the solution to others!

So, if Monty Hall asked you the above question, would you open the door you initially chose, or would you switch to the other one?

 

[VoidCat]

Counterintuitive? I don't even know enough about math to guess the wrong answer that's not counterintuitive.

 

[Debater Slayer]

Counterintuitive? I don't even know enough about math to guess the wrong answer that's not counterintuitive.

Well, setting aside any math, would you choose to open the door you had initially picked, or would you switch to the other door?

 

[VoidCat]

Well, setting aside any math, would you choose to open the door you had initially picked, or would you switch to the other door?

I can't drive. And I don't know how to care for a goat. It don't matter if I get a goat or a car I'm still getting something that I don't know what to do with. Might as well stick with the door i originally chose

 

[The Hammer]

Well, setting aside any math, would you choose to open the door you had initially picked, or would you switch to the other door?

It's, 50/50 afaict at that point, so I'd keep my original choice.

 

[Debater Slayer]

It's, 50/50 afaict at that point, so I'd keep my original choice.

Okay. I won't say yet whether this is the correct answer, but I can do it in spoiler tags if you want.

 

[The Hammer]

Okay. I won't say yet whether this is the correct answer, but I can do it in spoiler tags if you want.

Feel free to PM me if you wanna share the answer. I have no idea if I'm right lol

 

[amorphous_constellation]

I've read about that before, and I did not get it at first, and it is counterintuitive. Well the thing I'd like a simple, really simple breakdown of, is bayesian reasoning. If I understand a little more about that, then I'll feel like I accomplished something.

 

[Debater Slayer]

Feel free to PM me if you wanna share the answer. I have no idea if I'm right lol

Will do once I'm back on a computer!

 

[Debater Slayer]

I've read about that before, and I did not get it at first, and it is counterintuitive. Well the thing I'd like a simple, really simple breakdown of, is bayesian reasoning. If I understand a little more about that, then I'll feel like I accomplished something.

Sure! Would you like a general overview or a more in-depth explanation?

 

[amorphous_constellation]

Sure! Would you like a general overview or a more in-depth explanation?

As I understand, one can take a problem of likeliness, and plug it into an equation. Maybe take an example like that, and show me how you'd plug it into the equation

 

[We Never Know]

It's, 50/50 afaict at that point, so I'd keep my original choice.

I would agree here. Once one door is eliminated its out of the equation. The two doors left leave a 50/50.

 

[Subduction Zone]

I understand the Monty Hall Problem. I could also use a goat.

Spoiler

I am not going to switch because then my chances would remain two times out of three that I would win a goat. If I would switch the odds would be two out of three that I would win a car

 

[Heyo]

I would agree here. Once one door is eliminated its out of the equation. The two doors left leave a 50/50.

One should think so. After all the chance of tails in a coin flip is 0.5, no matter what the previous outcome was. The events are independent. After the goat was revealed it is a new experiment with two possible outcomes, each with a 0.5 chance.
That's where the error lies.

 

[Subduction Zone]

One should think so. After all the chance of tails in a coin flip is 0.5, no matter what the previous outcome was. The events are independent. After the goat was revealed it is a new experiment with two possible outcomes, each with a 0.5 chance.
That's where the error lies.

I disagree. I will not even put everything in the spoiler yet.

Spoiler

No. It is still the same experiment. I could try to give you the simple explanation why your odds of winning are 2/3 if you switch after the reveal of a goat

 

[Stevicus]

Starting with this thread, I'm planning to initiate a series of threads about counterintuitive problems in mathematics (or programming, if I encounter suitable ones), both for fun and as an opportunity to engage my passion for math.

The original Monty Hall problem is worded thus:



Monty Hall problem - Wikipedia

I posted the Wikipedia article to properly cite my source for the above wording, but since it contains the answer to the problem, I recommend not clicking the link before you have come up with your answer!

Before posting your answer, take your time to analyze it! This problem is an example of a veridical paradox, and it is so counterintuitive that Paul Erdős, one of the most prolific mathematicians of the 20th century, wasn't convinced of the correct solution until he saw computer simulations demonstrating its correctness.



Paradox - Wikipedia

Also, please only post your answer in spoiler tags so that you don't potentially reveal the solution to others!

So, if Monty Hall asked you the above question, would you open the door you initially chose, or would you switch to the other one?

I used to watch "Let's Make a Deal" as a kid. I thought the Monty Hall Problem was about trying to decide what costume to wear when appearing on the show and hoping to win the Big Deal.

 

[Heyo]

I disagree.

No, you don't.

 

[Shadow Wolf]

This is something I studied when I began learning more into gambling odds to sharpen my poker game.
I personally find it more impossible to explain to others than it is counterintuitive. Even after running a simulation to demonstrate to people this does actually work in the end I've still ran into people who just will not accept it works that way even if they don't understand why. The numbers are right in front of them but still they deny it.

 

[Subduction Zone]

This is something I studied when I began learning more into gambling odds to sharpen my poker game.
I personally find it more impossible to explain to others than it is counterintuitive. Even after running a simulation to demonstrate to people this does actually work in the end I've still ran into people who just will not accept it works that way even if they don't understand why. The numbers are right in front of them but still they deny it.

People often feel that they are investing in a choice and it can be hard to switch, even if the numbers tell them to.

 

[Shadow Wolf]

People often feel that they are investing in a choice and it can be hard to switch, even if the numbers tell them to.

It's not an actual choice though. Just something of some reason that has them not accepting how this works out.