I'm sure you've heard this one before. It goes something like this:
You are competing in a gameshow. Before you are three doors: behind one is a car, behind the other two are goats. You want the car (for some reason) and do not know where it is. The host asks you to guess which door the car is behind. After you make your guess, he opens one of the other two doors, revealing a goat, and offers this: you may have what is behind the other unopened door, or the one you chose originally. Which should you choose? Or more specifically: what is the probability that either option gives you of winning the car?
You've surely also heard that you should switch; the probability that the other door has the car is now 2/3. This has surely been explained to you in some way, and though you may not truly understand it, you probably accepted it as fact on account of it being explained as a "difficult" or "counterintuitive" problem.
I believe it's not that difficult. It just isn't necessarily true. The problem is that we do not know why the host has opened that door. Perhaps he only opens a door and asks you to switch if you chose the car at first, in which case you should never switch because it gives you a 0% chance of success. Okay, though it's a possibility, the problem is not written in a way that you would assume that is the case. Instead, you might assume that the host simply opened an unopened door at random. In this case, switching matters not: either door has a 50% chance of holding the prize.
This is the intuitive answer to the problem, and one you have surely heard is wrong. But it just comes from a different assumption about the problem itself. The common answer of 2/3 is correct when you assume that the host (knowing where the car is) will always open an unopened door that does not contain a car and offer a switch. However this is a little wordy to make clear, so it is not always included in descriptions of the problem.1
Thus, the problem with the Monty Hall problem. Just like how I always assumed "Hall" referred to a hallway of three doors and not the last name of Monty, many people assume a different version of the problem and correctly answer that one, but are told they are wrong. Explanations for the "correct" answer to the problem often do not realise that this is where things have gone wrong, and consist of various arguments for 2/3 being the answer that do not disambiguate the "monty picks a door at random"/"monty always reveals a goat" confusion.
Though it may be possible to convince someone of the truth of an incorrect statement, you cannot prove it. In mathematics, convincing arguments can be wrong, and proof is what truly matters. Nonetheless, your intuition can be more useful than you expect. It's a shame that the Monty Hall Problem often teaches quite the opposite lessons. Not to mention the demoralising effect of having someone attempt to convince you of what sounds like a simple mathematical fact that you just can't seem to understand...
I believe the problem with the Monty Hall Problem demonstrates something important about discussion more broadly. If you find yourself utterly unable to convince someone of something, even if you think it must be truly obvious by now, it's worth taking a step back: they may have simply assumed differently from you about some key fact. You can get much further if you can come to solid agreement about what it is you are actually discussing.
In fact, it may save everyone a lot of time if you could come to this agreement before beginning to argue at all.
1: Often it is specified in the problem that the host "knows what is behind each door". This does not actually help at all as it doesn't preclude the host behaving however he likes with that information. The only thing is does is hint at the "correct" assumptions. I think hinting is rubbish in a situation like this. Just make the actual problem clear!