Stumbling Intuition #1: The Monty Hall Problem

[muhammad_isa]

You can't win a car behind a door that does not open. There is no probability of his winning the car when the contestant is presented with the choice of the three doors because whatever door he picks, it is not going to be opened..

That's not true. He can choose a door and stick to it, and not switch.

Are we seeking the odds of his choosing the "right door"? or are we seeking the odds of his winning a car? Because they are not the same result, and therefor do not manifest the same odds.

We assume the contestant would prefer to win a car, and then they are "the same result".

In the scenario given, the odds to win a car is 2/3 if they switch.
..despite the fact that intuitively it might seem as only 50:50 at first glance.

 

[9-10ths_Penguin]

Not when that "extra information" doesn't correlate, though.

But it does.

It's just theater. Romoving one door from a blind scenario doesn't make the scenario any less blind. It simply limited the options. I still don't know anything more about the door Monty didn't pick than I know about the door I did pick. Because they could still be either door. The only way for me to know Anyang would be to extrapolate motive.

No, you don't need to extrapolate motive. You just need to remember the givens in the problem:

- the host knows where the car is.
- the host deliberately avoided revealing the car.

The odds being found on paper don't leave the paper. (Or the computer simulation.) They are only there because we are imposing a correlation at can only be imposed via the abstract properties of mathematics.

Your tangent seems like a very elaborate way for you to say "I don't understand the math here, so I'm going to assume that it can't be understood at all."

 

[PureX]

That's not true. He can choose a door and stick to it, and not switch.

That's not the proposition he's currently facing. You are assuming a choice that is not being offered and then basing your probability assessment on it.

We assume the contestant would prefer to win a car, and then they are "the same result".

They are not the same result when in the first proposition a chosen door is not an opened door.

In the scenario given, the odds to win a car is 2/3 if they switch.
..despite the fact that intuitively it might seem as only 50:50 at first glance.

Now you're just repeating yourself.

 

[muhammad_isa]

That's not the proposition he's currently facing. You are assuming a choice that is not being offered and then basing your probability assessment on it.

I don't understand.
What's not being offered?

 

[PureX]

But it does.


No, you don't need to extrapolate motive. You just need to remember the givens in the problem:

- the host knows where the car is.
- the host deliberately avoided revealing the car.

None of which reveals anything about the two remaining doors except that one has a car and one has a goat behind it.

Your tangent seems like a very elaborate way for you to say "I don't understand the math here, so I'm going to assume that it can't be understood at all."

You're avoiding the real issue, here. The fact remains that there is zero chance that the contestant will win a car when he chooses from among the three doors, because whatever door he chooses will not be opened. And he cannot win a car (or a goat) from behind a closed door. So the mathematical presumption that his first choice had a 1/3 probability of gaining the desired result (winning the car) is wrong. It had a 0/3 probability of gaining that result. The reason it's wrong is because the mathematician equated choosing the right door with winning the car. But these are not equal results at all. And in fact choosing the "right door" was not the result for which we are seeking to determine probability. Winning the car, was. And this makes the rest of this mathematical process for determining probability, wrong. Well, the mathematics isn't wrong, but the mathematical process has been wrongly applied, and so has rendered an inaccurate conclusion.

I like this scenario because it points up something that few of us ever consider regarding mathematics. And that is that it is an abstracted representation of reality. Like a language. And although it does afford us the very useful ability to correlate real information in ways that we could not otherwise comprehend, it can sometimes create illusory results, because in the end, it is an abstraction of reality, and not reality, itself.

 

[PureX]

I don't understand.
What's not being offered?

Winning the car or the goat is not what's being offered, because Monty is not going to open the door the contestant chooses. So all that's being offered to the contestant is choosing a closed door. It's just empty theater; unless there is a motive behind it that the contestant can determine and use to his advantage.

 

[muhammad_isa]

Winning the car or the goat is not what's being offered, because Monty is not going to open the door the contestant chooses.

Not straight away, no.
Monty will open one of the other doors first.
Then Monty will open the door you first chose, if you say so.

It is not a trick question.
..and Monty isn't shuffling the car and goats behind the scene.

 

[9-10ths_Penguin]

None of which reveals anything about the two remaining doors except that one has a car and one has a goat behind it.

That's not all it reveals if you think about it deeply enough.

 

[PureX]

Not straight away, no.
Monty will open one of the other doors first.
Then Monty will open the door you first chose, if you say so.
l
It is not a trick question.
..and Monty isn't shuffling the car and goats behind the scene.

The math has to accurately represent the reality as it is. And the reality is that the contestant cannot win the car by choosing ANY of the three doors. So mathematically, that part of the equation seeking to establish the probability of winning the car is 0/3, not 1/3.

 

[PureX]

That's not all it reveals if you think about it deeply enough.

Right. That's why above I included the whole scenario as viewed from the perspective of motive. Something the mathematician did not do.

 

[9-10ths_Penguin]

Right. That's why above I included the whole scenario as viewed from the perspective of motive. Something the mathematician did not do.

Again: motive is irrelevant.

 

[muhammad_isa]

That's not all it reveals if you think about it deeply enough.

That's right.
If the car is behind the door you first choose, then it reveals nothing.
..but if it is behind one of the other 2 doors, it does.
..and that's why it is 2/3.

 

[PureX]

Again: motive is irrelevant.

Unless it's not.

 

[Revoltingest]

I remember finding the Monty Hall thingie hard to
accept. A few random trials dint even support it.
But now it's obvious by looking at initial sets.
This shows 1/3 probability of the initial choice.
But 2/3 probability of switching after revealing
the goat.