The host knows which door the car is behind. Therefore if he makes decisions based on that knowledge those decisions can give you information about the location of the car. If there is some action that he is more likely to perform when the car is behind door 1 than when it isn't, and he performs that action, then you should increase the probability you assign to the car being behind door 1.

In particular if you've chosen door 2 then the host is more likely to reveal door 3 if the car is behind door 1 than if it isn't.

In the original problem, its presence _is_ relevant, as the car could be behind doors 1, 2, or 3. Say you pick door 1 - there's a 1/3 chance you are right.

The host is then left with doors 2 and 3. We know there is a 2/3 chance the car is behind _one_ of these doors. When the presenter reveals a goat (say in door 2), he is reveling information about this set of doors - there is still a 2/3 chance that the car is behind one of the doors in this set, but there's only one door left we don't know anything about (3). There is therefore a 2/3 chance that it is behind _this_ door.

The player are indifferent to switching under your scenario. Why not, I suppose?

2/3rds of the time, your initial choice will be a goat. In those cases the host deliberately avoids opening the remaining door with a car - thereby telling you where it is.

If you chose the car door initially, then indeed you get no new information. But that only happens 1/3rd of the time.

I am curious if that changes your opinion.

You know from the start that one out of three doors has a car behind it, and his opening one of the remaining doors doesn't change it. He just shows you what you already know.

When you select your door, you know there is a 1/3 chance that your door has a car behind it, and a 2/3 chance that it doesn't. The host opens another door, and there is still a 2/3 chance your door has no car. It has only been proven that the door the host opened has no car behind it.

EDIT: To clarify, I'll rephrase. When you select your door, you know it has a 1 in 3 chance of hiding the car, and importantly, you know that there is a 2 in 3 chance that the car is not behind your door.

When the host opens his door, you still know there is a 2 in 3 chance that your door doesn't have a car behind it, but are now guaranteed that, if you chose the wrong door in the first place, you will now choose the right door.

It's not that the host's door is eliminated from the possibilities, it's that you can either open your door or effectively open all the rest of the doors to find the car. Which would you choose?

"the host opens another door that he knows has a goat behind it, say No. 3,"

Using the knowledge of where the car is to avoid showing it, is more specific than just having the knowledge, and possibly still opening at random.

For the version of the problem where the host doesn't know where the car is ("I'll open one of the other doors, but I don't know what it will reveal"), it makes no difference.

For the version of the problem where the host knows where the car is and only opens another door if the car is behind the door you picked, you should obviously never change doors.

Usually when this problem is asked, the questioner (who usually doesn't understand the problem himself) doesn't specify which version of the problem he's asking about -- all of the cases I know where a famous smart guy has answered "incorrectly" come down to the problem not being described to them in precise terms and them not making the same assumptions about the host's behaviour as the questioner.

It doesn’t matter that the host meant to do that. Now that the information is revealed, the probabilities have changed.

Obviously if the host revealed the car, that would also affect the probabilities!

I can make sense of it by drawing out the probability trees and see that it's 1/2 rather than 2/3 in this case, but it actually sticks with me if I think more like so:

My prior is that there's a 1/3 chance I picked the car; that's the world where the host can't reveal anything other than a goat.

In the scenario where the host then reveals a goat intentionally, my prior doesn't change. The host can always reveal a goat. This gives me no information. Therefore, it's still 1/3 that I picked the car, so the remaining door must be 2/3, so I switch and get a 2/3 chance.

But in the scenario where the host reveals a door at random, and it happens to be a goat, it's time to update my priors. Since he didn't accidentally reveal the prize, the probability that I'm in the world where he couldn't accidentally reveal the prize is increased relative to the probability that I'm in the world where he could have.

Imagine a universe with many simultaneous Monty Hall clones playing at once in many studios, where Monty doesn't know and opens another door at random. If that door has the car behind it, Monty and contestant are both shot in the head and the studio burned down and erased from all records. This bloody culling of branches of the probabilities is the same as effected by giving Monty the knowledge and telling him to act on it.

Of course it matters.

The host uses his knowledge to AVOID opening the door with the car (and thus ending the game prematurely).

So he essentially gives out this information: "the door with the car is one that I, the host, haven't opened".