> The thing that is often de-emphasised in the presentation of the problem, in order to make it seem more mysteriously paradoxical, is that the presenter knows where the car is and this knowledge is always used perfectly. If the question always ended with "remember: Monty knows where the car is and will use this information", it would be more obvious.

The associated line of reasoning resolved the paradox for me. If I stick with my original choice, it is as if I ignored the new information. If I switch the choice, I react to the new information.

The extreme of this is to pick among countably infinite doors, having the presenter open countably infinite doors and leaving just yours and another closed. Who could reasonable suggest that the chance is still 50/50, assuming you don't flip a coin and base your choice on that?

> If I switch the choice, I react to the new information

This assumes that him opening a door is new information. It isn't unless you make an assumption about the game masters intentions.

If the host doesn't want you to win then you will lose 100% of the time if you switch, if the host wants you to win then you could win 100% of the time by switching etc. But people think that these assumptions are "obvious" so of course you must make them, but that makes the riddle bad.

If the riddle doesn't say that the host always opens an empty door after you pick a door, then the typical solution isn't correct. People does the typical "X happened once" and assumes it means "X always happens", that is a typical naive assumption but no, just because it happened once doesn't mean it always happens, you can't make that assumption here unless stated in the riddle.

>if the host wants you to win then you could win 100% of the time by switching etc.

66% of the time. 33% of the time you picked right first time and lost it by switching. The host never conveys "dont switch."

> If the question always ended with "remember: Monty knows where the car is and will use this information", it would be more obvious.

And perhaps also, “remember: Monty will always open a door, and the contestant knows it”.

Makes me wonder if there were similar shows where the host can choose not to open a door.

That presumably quickly gets into a game-theoretic double-triple-etc-bluff affair where you have to assume the host is using his knowledge against you.

If you choose the door with the car, the host will open another door to tempt you, so then you don't switch, unlike in traditional Monty Hall. And if he didn't open a door, well, that means you choose a goat, so you should switch for a 50% chance. Except the host knows you're thinking this. And you know he knows. And he knows you know he knows. And...

Perhaps, iterated infinitely, this eventually resolves into a limiting expected value for switching and not, but I have no idea how to compute such a scenario.

See: https://en.wikipedia.org/wiki/Monty_Hall_problem#Other_host_...

The puzzle is is commonly stated is not how any TV show ever worked. In a real (repeated) TV show the host behavior will be non-deterministic, and will sometimes be benign, sometimes adverse, to increase the suspense and ratings. If the host follows a deterministic set of rules (as is usually implicitly assumed in the puzzle version), the optimal strategy is pretty easy to work out.

> If the host follows a deterministic set of rules (as is usually implicitly assumed in the puzzle version), the optimal strategy is pretty easy to work out.

And yet it is the subject of endless discussions. The optimum strategy is pretty easy to work out, yet not widely believed.

You can browser the countless HN or even Reddit threads and it's always filled immediately with one of the few standard explanations (which are all in the Monty Hall Wikipedia article). None of the "endless discussions" actually disputes the strategy or nobody needs to be convinced. It's all about whoever things they can come up with the most "intuitive" explanation. At this point I'm starting to thing that it's more a myth than reality that it's "not widely believed".

In the scenario where they are shot that you described, it is still better to switch!

You can’t switch if your universe has been pruned Morty.

This explained it better to me than the article

wow... what a grim approach!

That Escalated Quickly would be a great name for a game show.

Next on 4: Brian Cox presents Quantum Squid Game, in partnership with Academi.

The problem I always struggle with is the premise that presenter not only knows which box is the prize, but also does not want you to win. In other words, there should never be a scenario where the show host gives you an opportunity to switch boxes unless you have already chosen the prize box, in which your choice should be to not change boxes. I guess I just naturally assumed that game shows don't want contestants to win and the odds are against you.

> there should never be a scenario where the show host gives you an opportunity to switch boxes unless you have already chosen the prize box

The host always gives you an opportunity to switch boxes.

That is an assumption though, the problem usually doesn't say that.

What would the problem even be asking if you don't get that choice? The problem (as given) requires you to make a choice about keeping or switching—if the host doesn't offer you that option, then what is the problem even about?

I'm giving you an up vote for creative writing and nothing else. I'm not claiming anything about your data, just your creative writing skills. Thank you.

I agree that this additional information is necessary to understand the problem. I've seen that show maybe twice in my lifetime, both times when I was a little kid. I did not know that Monty never opens the door with the prize behind it. Not one time in the history of the show did he ever open the prize door. Knowing that is additional information, and can be used to figure out the problem. But you need to watch the show more than twice to know that.