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You're in a game show. There are three closed chests, 1, 2, and 3. One of them contains a prize, the other two are empty. You can choose one chest and choose chest 1. "Are you sure?" the game host asks. "You can still switch if you want to!"
At that very moment, chest 2 malfunctions and accidentally opens. It is empty. "Huh," says the host, genuinely surprised. "Well, my offer stands."
What should you do?
Most people here know the Monty Hall problem. But, crucially, this was not the Monty Hall problem! The 31% of the people who said that it doesn't matter if you switch it or not were right.
Of course, this being a Monty Hall-like problem, I don't expect people to just believe me. So I've written a long blog post to explain:
http://lilith.cc/~victor/dagboek/index.php/2024/10/11/monty-hall-troubles/
@victorgijsbers I didn't fully believe this principle until I simulated it
@no2nsense I wrote a reply, but I decided to wait until a few more people had a chance to fill in the poll!
@victorgijsbers i see, in this case the possibility space includes the price having been in the accidentally opened one. Tricky!
@no2nsense @victorgijsbers I still remember in high school writing a simulation on my graphical calculator and even after that I still didn't entirely believe it.
@victorgijsbers I find it terribly difficult to wrap my head around this one. Certainly not the kind of conundrum where afterwards you cannot see anymore what tripped you up originally.
@victorgijsbers I don't know if it doesn't matter or not, but staying with my choice defenetly won't increase my chance. So I'll switch. (Does this tell I'm an engineer, not a mathematician?)
@technetium @victorgijsbers This is definitely one of the most original “intuition-based” arguments I ever read in this classical puzzle
@victorgijsbers To help myself visualize the problem, I wrote out the alternative scenario.
At that very moment, chest 2 malfunctions and accidentally opens. It has the prize. "Huh," says the host, genuinely surprised. "This seems too easy. How about a quick round of trivia and then you can have it?"
Possible scenarios:
- Prize is in box 1, box 2 opens, host says "my offer stands."
- Prize is in box 2, box 2 opens, host says "this seems too easy." (This didn't happen.)
- Prize is in box 3, box 2 opens, host says "my offer stands."
This differs from the scenarios in the classic Monty Hall Problem:
- Prize is in box 1, host opens box 2.
- Prize is in box 2, host opens box 3.
- Prize is in box 3, host opens box 2.
In the classic version, scenario 2 and scenario 3 add together: 1/3 + 1/3 = 2/3. In this version, scenario 2 is ruled out, adding nothing.
@victorgijsbers Though Monty Hall pointed out some other flaws in the problem. He was under no requirements to follow any particular rules; if you picked wrong he could just open that door and show that you won nothing.
Sometimes he'd offer money for people to switch (or not switch), potentially tricking them into sticking with their wrong answer. Looking back, he commented that most contestants would have been better off taking the guaranteed money, rather than trying to outsmart the guy who'd been doing this for years.
depends...
1) do you think the host wants you to win?
2) do you think the host thinks you know the monty-hall problem?
Jeroen 
@victorgijsbers You're in a game show. There are 1000 closed chests, numbered 1, 2, 3 and so on. One of them contains a prize, the other 999 are empty. You can choose one chest and choose chest 1. "Are you sure?" the game host asks. "You can still switch if you want to!" At that very moment, 998 chests malfunction and accidentally open. They are all empty. "Huh," says the host, genuinely surprised. "Well, my offer stands."
What should you do?
@jeroen @victorgijsbers this is my favourite intuition pump for the Monty Hall
@tomstafford @jeroen And what's your intuition in my riddle?
@jeroen @victorgijsbers
This is exactly the way I explain it (well, either 1000 or 100 which is clear enough to me). It seems so obvious when the number of chests/doors is high: was I really so lucky in my first choice? Probably not!
@agaitaarino @victorgijsbers Yeah, indeed. This really made it intuitive to me as well.
@victorgijsbers I answered “switch to 3”, but now I wonder if the fact matters that it’s a random chest that malfunctions and just happens to be the empty one and not a deliberate choice to open the chest known to be empty.
@jeroen @victorgijsbers
I'd say the argument is actually more clear and solid in this random/accidental version of the problem. If the host chooses to open a chest/door you cannot know whether the director of the show instructed them to do it only (or more often) when the person chose the right one initially, and it could then be just a trick to throw you off.
@agaitaarino @victorgijsbers The fundamental difference is that opening a random door will sometimes reveal the prize. With the original Monty Hall scenario this NEVER happens. When you have 1000 doors, opening 998 random ones has a VERY high chance of revealing the prize. If, through chance, none of the randomly opened doors reveal the prize you still don't have any extra information based on which you should decide to switch. Your new chance of winning becomes 50%, whether you switch or not.
@victorgijsbers Devious. That’s one way to resolve the purported ambiguity.
@victorgijsbers would be funny if the results would be 66%/33%/0% 
@victorgijsbers I spent some time thinking about whether the host was surprised at the chest opening or at it being empty. This complicates matters a bit. 
@victorgijsbers Well, I know that in the original setup, I should switch. Because there’s a 2/3 chance that the prize is in box 3. Now, without racking my brain over if the odds are 1:2 or 1:1, switching either increases my chances or leaves them as they were. Since there is no harm in switching (it will never leave me with less than 50% chance), I’ll switch.
@victorgijsbers
I'd take chest 2. I want the uncertainty to go away as fast as possible.
@victorgijsbers If the host is indeed genuinely surprised, he must have thought the prize was in chest 2, so he doesn't have knowledge in which of the other chests the prize is in, and if his intention were to manipulate you either way, he has no way to. The probability of the prize being in either chest 1 or 3 is 1/2, so it doesn't matter if you switch or not.
@victorgijsbers Ok so now to reap our community respect being a 31%er. Yes yes, bring it on!
And thank you Victor for sharing this cool variation of the game show and the explanation.
@victorgijsbers you just boosted my self confidence
@victorgijsbers Very clever, I didn't get this until I read the boxers variant.
@victorgijsbers Dank voor ‘t raadsel en de uitleg!
@victorgijsbers I can’t help but think of LLMs here, how we criticize them for using vague statistics, but people also turn to hypercorrection because they learned to answer a certain way without really internalizing why. I wonder how many of the 31% understood why they were right, whether they would fall for the original.
Minor correction: it should be “00 to 99” at the end. As posed, there are 81 ways to pick 2[^2] and [^2]2, but only 80 to pick 22 and [^2][^2].
@jonathanavt You're right, I'll fix the 00 when I'm back behind my PC!
@victorgijsbers although I’m in team “doesn’t matter”, there *is* a wrinkle. After the accidental opening the host says “my offer stands”: does that imply he could have changed his mind at that point? If so, reasoning about *that* choice is quite a bit trickier.
@victorgijsbers if he retracts the offer, you might think you know for sure where the prize is (and that you’re not getting it). But I think his best strategy involves some randomisation so that if he *doesn’t* retract the offer, you don’t thereby learn your best choice. And if so, that opens the door for the kind of bounded-rationality prior-bending you get from things like “does he know whether you know the Monty Hall problem?” It gets real messy real fast.
@victorgijsbers I am by no means an expert, but one thing I feel many don't fully grasp is that probabilities only have meaning when applied to populations. Saying a person who can choose only once has a 33% chance of choosing the right door is in many ways useless. It has no predictive value for the outcome of that single choice. They either choose right, or wrong. The 33% becomes meaningful ONLY when considering many attempts at choosing.
@jeroen Well, that's a possible interpretation of probability, but by no means uncontroversial! One could certainly think that probabilities are meaningful in contexts where we don't talk about populations.
If you and I play a game where we both make up a nonsense word, and you win if we make up the same nonsense word, and I win otherwise, then even if we play that game only once and nobody in the universe ever plays it again, surely it is true that you have a very small probability of winning.
@jeroen Or a very different example, someone could say: "Given the evidence, it is very unlikely that there are any undiscovered planets between Pluto and the Sun." And that might seem a perfectly meaningful use of probability, even though there is no population of alternative solar systems (I'm not even sure what that would mean).
@victorgijsbers Of course! Probabilities are very _useful_, when used appropriately. My point is that probability is a surprisingly subtle concept, and is often used in a hand-wavy, informal kind of way. When the weather report says there's a 90% chance of rain tomorrow, that's very useful to me, and I'll definitely pack an umbrella. But when you sit down and think about that statement, what does is tell me? _Will_ it rain tomorrow? Maybe, maybe not.
@victorgijsbers All it says is something like "given 100 days with conditions like today, it will rain the next day approximately 90 times". Probabilities are statements about populations, they cannot predict individual outcomes. In many ways, probability theory is a formalized way of dealing with incomplete information. Flipping a coin has a 50% chance of landing heads or tails. Unless we control the coin toss precisely enough, in which case we can perfectly predict the outcome.
@victorgijsbers The same is true for any other example you can think of. Predicting the weather? The probabilities we hear are caused by the limited resolution of the measurement of today's conditions. Will a vaccine prevent illness in one individual? We don't understand the human immune system well enough for that, nor the other factors that determine the outcome, so we have to make do with looking at efficacy in larger populations.
@victorgijsbers @modulux haha this sort of hypathetical was presented in the film 21. Also in Robert J Sawyer's novel Wake. I recognise this. very cool
@bermudianbrit @victorgijsbers I got it wrong. I thought it was the Monty Hall problem but after reading the thread I was convinced that it was different.