I need to write a blog post or something convincing everyone we need to stop talking about the Monty Hall problem and replace it with a new problem with all the ambiguities removed. (Unless ambiguity is the point, then Monty Hall is fine.)
For example, going by that ddg search, one result is making a fuss about not knowing whether Monty opens a door at random and happens to show a goat, or purposefully opens a door with a goat behind it. But we do know: it's always on purpose, Monty never opened a door with a car behind it, thus prematurely ending the bet. So there's no ambiguity.
The problem is cool because the right answer doesn't seem intuitively right, even though it can be formally shown to be right.
There's probably a name for that cognitive bias, but I don't know it.
We only know that if the problem tells us. Sometimes it doesn't.
> There are no ambiguities in the Monty Hall problem
The problem has been written up thousands of times. I'm sure that some writeups are sufficiently unambiguous, but many are not. For example, consider the two "variants" described by this comment https://news.ycombinator.com/item?id=8664550
> The host selects one of the doors with a goat from the remaining two doors, and opens it.
> The host chooses one of the remaining two doors at random and opens it, showing a goat.
This commenter was trying hard for semantic precision, and yet, I think if you encountered the first variant in isolation it would be perfectly reasonable to interpret it as "The host [randomly] selects one of the doors with a goat [although he might have selected the prize]" even though this is clearly not what the commenter was attempting. If you disagree, that only proves my point: this problem is prone to silly and wasteful semantic debate, rather than the interesting probability result it should be focused on.
I think there's a good argument that the intended interpretation should be the favored one. If he doesn't use the knowledge to always reveal a goat, why did they bother specifying that he has the knowledge?
But it still doesn't quite explicitly say that he's not picking randomly.
Notice if Monty uses the knowledge of whether to open the door based on what you choose, he'd be giving the whole game away, so that's not it.
It's not at all about "rules lawyering" the premise of the puzzle.
It's like going around saying that "counterintuitive answer to equation x/x=1 is 17" and then "prove it" by dividing 17 by 17 . When confronted with the fact that in math to solve an equation is to find ALL solutions and not just one, you say "It's not at all about rules lawyering the premise of the puzzle". Well avoid dealing with the math then because the math in fact is all about the rules.
The answer IS mathematically correct. It's just counterintuitive and it made some PhDs trip.
Nobody here -- not even you, before -- was arguing it's mathematically incorrect. Some people, when told the right answer, claimed the problem is underspecified and admits more than one context that may change the answer, which is not at all the same as saying the answer is mathematically incorrect! Not even Diaconis, the person some of you are so eager to defend (for some bizarre reason) claims it's "mathematically incorrect"!
It seems you're grasping at straws now.
It's not different from how in middle school you can incorrectly assume that "x is positive" when solving x^2=4 in R. An answer "x=2" is mathematically incorrect.
Well, you're wrong. What's worse, the followup by vos Savant clarified this.
> In probability theory you can't assume independence of two random variables unless it's a part of the problem statement
In most logic puzzles you can safely assume an interpretation of the problem that makes sense and which doesn't go into extraneous tangents. Going "well, akshually, if we assume a spherical goat..." is usually a bad sign.
Frankly, all of this still reads as an a posteriori rationalization for finding the solution to the straightforward formulation of the puzzle counterintuitive.
Luckily, when I was introduced to that problem many years ago it was presented correctly and the answer albeit counterintuitive was perfectly clear. Developing an intuition for it was also rather easy (what is the chance I guessed incorrectly at a first try? it's the answer).
What's above is my reaction to the incomplete formulation of the problem and an incorrect answer that follows.
The reason I'm so annoyed by this is because probability theory is very fragile and only works when applied with absolute precision. If you follow your approach with the Two Envelopes Problem and make some "reasonable assumptions", you get a crazy answer (always switch). And people who are in the business of logic puzzles rather than probability theory wouldn't even know the difference.
Therefore I would rather discourage people from working on logic puzzles and suggest doing the actual math instead.
The correct solution is to protect one of the options and then choose the other option.
The real challenge is coming up with appropriate flavor text for this idea, but I think I'll get it eventually.