[logicallee]

The issue is that a lot of intuitive stuff is wrong. When you formalize, you remove the simple, intuitive explanation - but you make it much harder for you to remain wrong, if you are wrong - or to become wrong, if you started off right.

As a simple explanation, consider the difference between explaining the Monty Hall problem - which might seem to be philosophical, open to interpretation - and coding it up. The moment you code it up --- go ahead, code up a monte carlo simulation that compares the two alternatives of switching doors or maintaining the same choices, and spits out a running count of which is what percentage correct. I'll wait while you code. ---

the moment you do that, you see two lines in your code that make the explanation 100% irrefutable and completely obvious.

That is why papers are written this way. Intuition can go both ways.

[SilasX]

That would be more convincing if the scientific papers were written in a way that make the point as clearly as the coded-up version of Monty Hall problem. In practice, it's more like they publish the assembly code and when you ask why they didn't do it in Python or something, they lecture you about the need for formal rigor.

[tel]

I think there's generally selection bias about what part of an exposition makes the "a-ha" hit in two ways. First, your a-ha moment may not be the same as someone else's, but you're less likely to observe theirs. Second, your own a-ha is likely the product of a larger production than the moment itself of which you're most attuned to.

A good mathematical author must be guarding against both of these selection biases.

[johnbm]

Nah, Monty Hall is trivial to demonstrate. Just do it with 100 doors instead of 3. Problem solved, intuition remains.

[Crito]

Such that the contestant chooses 1 door, the host then opens 98, and they are given the opportunity to switch to the last remaining door?

That's actually a pretty brilliant way of explaining it. With numbers like that the answer becomes much more intuitive.

[logicallee]

Why 100? Why not just 5? Some people would 'get' it at five, some people at 100, and some people at a million. If you have to choose out of a million doors, and no matter what the host opens all but one of them, so that your prize is either behind the door you picked, or behind the other one -- then should you switch your choice?

Well, obviously, you should - with a million doors, it becomes obvious that you have just a 1 in 1,000,000 chance of having picked it.

But thing is - that "obvious" thing 'should' be just as obvious with 1000 doors, 100, 20, 5, or...3....

It's a matter of degree - not kind.

So appealing to a way of intuiting it that is a lot more 'obvious' - while in fact having the exact same format of question, just goes to underscore how fickle intuition can be.

That said, taking individual variables to ridiculous extremes is a great way to thought experiment and an awesome way to get intuition to work better.

[Crito]

Yeah, I'm not wed to a particular number... I'm just observing that larger numbers seem to make it more intuitive than, say, 3 doors.

[ColinWright]

I never understood this assertion. Most people I explain it to in this way still think it's 50:50 because you only have two doors left.

[Shamanmuni]

That's odd, I found the 100 doors example is the most efective way of explaining it.

It's easy, pick a door, then the host discards 98 doors in which the car isn't. Do they still think that the probability of the other door left is the same than the one they picked? I want to play gambling games with them!

[ColinWright]

So people I ask say this:

    Pick a door, the host discards 98 doors
    where the car isn't.  There are now two
    doors left, so it's 50:50.

Actually, I'm with them (except for the 50:50 bit). I don't find the 100 door version any more convincing than the 3 door version. Under the usual assumptions the reveal of the other doors gives no information about the one you picked, so that will always remain 1/N. The remaining door will therefore be (N-1)/N, which is bigger if N>2. So switch.

[johnbm]

Because it is near impossible to open 98 doors at random and not reveal the car.

[ColinWright]

To the people who have trouble with this, that's neither relevant nor helpful. They just say "two doors left, hence 50:50."