[I_Am_Nous]

The Monty Hall Problem is a fun one because you can try to approach it from a purely analytical perspective and get one answer, while incorporating the whole situation (especially the fact that the final probability is not natural as they force the final decision into far fewer doors than originally present) and testing you can find a different answer entirely.

I suppose this is an interesting corollary with discoveries made by deep theoretical mathematics. While something may seem possible because "the math checks out" it could be only theoretically possible as it relies on some unnatural value to "be" possible in the first place.

Testing is where hopeful theories are smashed by reality until all that remains is the verifiable truth. Truly, why wouldn't we test?

[naniwaduni]

Fundamentally, the trouble with the Monty Hall problem isn't that analysis comes to the wrong answer, it's that people often come to the wrong model when reasoning about it informally.

It's not any harder to do the "correct" analysis than to write up a simulation. It's mostly just easier to convince yourself that the simulation matches the problem description when it reaches the unintuitive result.

[s1artibartfast]

Fundamentally, I think the real trouble with the Monty Hall problem is that the assumptions of the game are not clearly stated. Because of this, people come up with different models.

[alexdowad]

That's absolutely right; further, if you explicitly model the behavior of the game show host, you can exhibit models under which "it's better to switch" and models under which "it doesn't matter if you switch or not".

[vince3455]

>models under which "it doesn't matter if you switch or not".

Could you provide an example? It seems obvious that a switcher wins exactly when a non switcher looses, which is 2 / 3 ?

[alexdowad]

Take a game show host who lets you choose a door, randomly reveals what is behind one other door, and then gives you an opportunity to change your choice. This game show host CAN (randomly) reveal the prize; he has equal probability of revealing ANY of the unchosen doors.

Say you are playing the Monty Hall game with this host. You choose your door, he opens another door, and it happens (purely by chance) that there is no prize there. Do you still believe that you have a 2/3 chance of winning if you switch to the other unopened door?

[zaccusl]

Isn't that a different problem entirely? The original is that the host reveals a door without the prize.

Aren't you modeling an entirely different problem as opposed to modeling the same problem with a different model, since the problem states the parameters and you are changing those?

[Natsu]

This modelling ambiguity is resolved by do calculus, which makes a clear distinction between intervention and observation: https://arxiv.org/pdf/1305.5506.pdf

[pmontra]

That's how it went when I was solving problems at the Statistics course at university. I modeled the problem perfectly, got the wrong result. Changed assumptions, got the wrong result. Checked the solution, its reasoning didn't make much sense anyway. Run a simulation, got an approximate result close to the correct solution.

[dotancohen]

This sounds like the classic "tweak the model until the results fit with our preexisting conclusion". Very common across all industries unfortunately.

[zh3]

Also known in a derogatory fashion [0] as "adding epicycles" (after the Ptolemaic view of the heavens).

[0] https://en.wikipedia.org/wiki/Deferent_and_epicycle#Bad_scie...