[RedDevil]

The thing that makes it hard to get right is that it looks like a lottery.
I won a car in a lottery; the Hyundai I10 in my garage, actually. I know how it feels.

But the contest is not a lottery.
The host can pick a door with a goat because there will always be one door with a goat left after your first choice.

I can break it down as simple as this:
The chance that your first choice is right is 1/3.
The chance that the car is behind one of the other two doors, of which one will be opened to reveal a goat, is 2/3.

The two doors you did not choose still have a combined 2/3 chance of having the car behind them. But when you know which of those two does not have the car, you know the other has the full 2/3 chance.

Got it? No?
Let's change the experiment:

Running the contest with 1.000.000 doors
1 car, 1 million doors.
We send the goats on leave this time.

You pick one door at random. You have 0.0001 % chance of picking the car.
There is a 99.9999 % chance that the car is behind one of the other doors.
There are (at least) 999.998 remaining doors that do not have a car.
Now it comes:
The host opens 999.998 doors, carefully skipping the one door hiding the car (unless you've had the uncanny 0.0001% chance chance of picking the right one at first, in which case he skips a random door)

Will you swap or not?

You should.
Going to high numbers clearly shows how selective the host operates.
With every door he opens he does not increase the chance that your original choice was right, though it may feel that way.

Because unlike a lottery, he does not pick doors at random.
He opens only doors that have no car.

Running a lottery by contrast
If it were a lottery, initially you´d have a 0.0001 % chance that a door reveals a car for each door, just like the contest.

If after half the doors were opened still no car was revealed, then you´d have a 0.00002% chance that any remaining door would have a car.
But you also would have a 50% chance of being out of competition already.

After 999.998 doors and no car shown the remaining two doors, the one you´ve originally chosen and the remaining one, each have a 50% chance of hiding the elusive car.
But you´d only have a 0.0002% chance of getting that far. Over the whole the chance that those particular two doors have the car is still just 0.0001% each.

In the game there is no element of chance after your initial choice.
You have 100% chance to make it to the last 2 doors because the host will not pick the door with the car.
Nothing alters the chance that you did pick right first time.