Another Monty Hall thread

No, I’m not asking about the math. I understand the odds. This is a different topic.

Part of the set-up is that Monty knows what’s behind each door. Has anyone ever discussed the part of the problem? Did Monty already know before the contestant made their choice? If he did, did he have a tell? It occurs to me that you could watch past episodes of Let’s Make a Deal and observe Monty’s reactions. You might discover, for example, that Monty always raised his right eyebrow when you picked the wrong door or frowned when you picked the right door.

If the studio was concerned that Monty might be inadvertently tipping off the answers, they might have kept him in ignorance along with the contestant when the choice was made and then signaled to Monty which door he should open.

Another issue was whether the choice of the door with the prize was truly random. Did anybody ever keep records? Maybe the studio felt that the show was more dramatic when Monty opened the middle door so that door was left empty more often than the right or left doors.

Everyone treats this as an idealized math problem but I’m wondering how much the actual show differed from the idealized conditions.

On the show he knew what was behind the doors and tried to help the contestants.

From the wiki for Monty Hall:

" Hall gave an explanation of the solution to that problem in an interview with The New York Times reporter John Tierney in 1991.[28] In the article, Hall pointed out that because he had control over the way the game progressed, playing on the psychology of the contestant, the theoretical solution did not apply to the show’s actual gameplay."

There are some games where, even if Monty knew things in advance, he couldn’t have shown a tell that would have meant anything; the “old-fashioned cash register” game comes to mind.

The closest I noticed to a “tell” was, if, right before a couple was offered anything, Monty made it a point to actually take the contestant’s “traded item” (technically, they’re trading things for the prizes, but they don’t always actually take away the contestant’s items), then the item would end up being in a box or behind a curtain later in the same deal.

That’s the kind of thinking that throws off the odds. Hall might have thought he was playing on the psychology of the contestant but he was actually introducing his own psychology into the game.

It’s like asking people to pick a random number. If a group of actual people pick supposedly random numbers, they do not produce a true random assortment. We naturally fall into patterns - even if we’re not aware of the pattern.

Just a reminder, the problem isn’t the same as the game show. Monty didn’t offer you a chance to switch. The original problem was just posed as “a game show.”

The original Marilyn Vos Savant column that started the furore is here, or at least a follow-up column that quotes the original one.

She was responding to a question from a reader written in to her newspaper column, not from any direct knowledge of an actual game show in practice. The person who wrote to her did not mention Monty Hall, and the question did not really give sufficient information about the rules of the hypothetical problem he was asking about. And I think vos Savant did mess up in failing to clarify that in order for her solution to be valid, it’s clearly necessary to establish that the rules require that the host does not have any discretion - he must know what is behind each door, and he must be bound to proceed mechanically to always reveal a goat in every iteration of the game, whatever the initial choice of the contestant.

However, even given that lack of clarity, she was clearly far “more” correct that all the geniuses writing in afterwards to insist that the probability is 0.5. There is no simple form of that game that I can conceive of that always offers a switch and where the solution is 0.5, where it makes no difference whether you switch.

Of course, if the host has discretion and does not follow a strict set of rules, then no purely probabilistic solution exists.

I realize the Monty Hall Problem is distinct from the actual television show Let’s Make a Deal. I said right in the OP that my question is how much the television show differs from the math problem.

But from the phrasing of the last paragraph of your OP, what you didn’t seem to be clear on is that the vos Savant column that started the furore never claimed to be a solution to any real-life game show, and did not mention Monty Hall.

Well, it was his show and they definitely wanted to end those shows with a winner.

I’m not really sure what point you’re making here. I didn’t mention Marilyn vos Savant in the OP. So why are you bringing her into this? It’s not that I’m unclear on the issue, it’s that I don’t see the relevance.

Your OP (especially the last paragraph) reads as though you think the well-known solution was intended as an attempt to reduce the actual game show to idealized conditions. Pointing out that it wasn’t does not seem irrelevant to me.

Then let me clear it up. That was not my intent.

I looked at the WIKI for this.
There is a table with three rows of possible car placement.
But there are four scenarios. If your pick is wrong, then Monty can only reveal one door. If you pick right, he can choose one of two doors. So there are four scenarios to make the choice to switch or not.
It seems to work out 50/50 in that case.

Maybe I am making a mistake. But I did build a scenario table, for each case of car placement. Each case of initial pick. But included the fourth scenario where Monty picks a different door to reveal, in the case that the car was initially picked.

Not sure if including all possible scenarios does in fact change the real outcome. But it looks that way in thee table.

One can also bring into dispute whether Marilyn vos Savant can be considered to be the ultimate authority on everything. Sure, IIRC it was claimed in her column that she was the world’s smartest person. But we here at the Dope know otherwise.

I think you’re missing the probabilities. The scenarios you describe are not equally likely to occur.

There are three doors to start with and only one has a prize behind it (the “right” door). This means you have a one-out-of-three chance of picking the right door and a two-out-of-three chance of picking a wrong door.

So the true probabilities are this:

1a - You pick the right door and Monty opens wrong door #1. (16.6% chance)
1b. -You pick the right door and Monty opens wrong door #2. (16.6% chance)

2 - You pick wrong door #1 and Monty opens wrong door #2. (33.3% chance)

3 - You pick wrong door #2 and Monty opens wrong door #1. (33.3% chance)

Right. I was getting a fuzzy intuitive concept of having to include more players and scenarios. But the first level of probabilities only lets 33.3% of players face either 1a or 1b.

All right…one more time. Disclaimer: I don’t do cults of personality, so I really don’t give a damn about Marylin Vos Savant or “Cecil Adams” (who I’m now completely convinced doesn’t actually exist, but I guess we’ll never know now, huh? :roll_eyes:) and have zero interest in asinine arguments over “world’s smartest” anything.

The question as originally posed was if it was more advantageous to stay on the current door or switch to the last remaining door given that the host opened one of the losing doors beforehand and did not have any choice not to do so. Given all this, the correct move was to switch, because if you stuck you won if you guessed correctly, which had a 1/3 chance, whereas if you switched you won if you chose either losing door, of which there was a 2/3 chance. That is the answer to that problem.

Now, if you’re going to argue that any real game show host would never be so stupid etc. etc. (which “Cecil” actually argued in his column) that’s a completely different problem. Which is perfectly fine provided that YOU MAKE IT CLEAR THAT YOU ARE CREATING AND SOLVING A DIFFERENT PROBLEM. A variant, if you will. The problem is that different solvers keep making all these different rules and conditions and situations and act like it’s the original problem, with the result being that everyone can’t even agree on what the hell they’re discussing.

It’s as if there was a “knights and knaves” thread, but instead of presenting a variety of problems, everyone got into a massive argument about the “correct” solution to “the knights and knaves question”, and it all ended in a huge ball of confusion because no two people were talking about the same thing.

Dammit, this could’ve been fun. :slightly_frowning_face:

Little Nemo - You seem genuinely curious, so I’m game. Present your variant in full and I’ll give it a shot.

I wrote “This is a different topic.” in the first line of the OP. If people couldn’t figure out this was a different topic after that, I don’t feel that’s on me.

I’m not presenting a variant. Again, I explained this in the OP; “Everyone treats this as an idealized math problem but I’m wondering how much the actual show differed from the idealized conditions.”

Incorrect. The question as originally posed did not specify that the host had no choice of whether to open the door. The fact that that was not, in fact, specified is the root cause of at least half of the debate that has gone on about this question, and the fact that vos Savant did not make it clear was simple negligence on her part.

And “Cecil Adams doesn’t actually exist”? So, all of those columns just materialized themselves out of the ether? Someone wrote them, and the someone who wrote them uses the name “Cecil Adams”. Which, yes, is a pseudonym, but so what? I’ve never heard anyone claim “Mark Twain never actually existed”.

Maybe that’s the way we can put an end to these arguments. The next time somebody raises the Monty Hall problem, we’ll point out that Monty Hall never existed. (His actual name was Monte Halparin.)