The Monty Hall "Paradox" explained

Most of you might be familiar with this so called paradox, the Monty Hall problem, and many of you probably know the "counter intuitive" solution to it. Some might have just heard the solution and accepted it, others might have pondered it but never really gotten why it is true. I don't think this is because the problem is difficult, but rather because it is always framed badly as this brain teaser, that will spark debate even among statisticians. So i will try to expand the problem to simplify the solution.

For those who do not know the problem, here it is.

You are on a game show. In front of you is three doors, one with a car and two with goats behind them. You do not know which doors contain what, and you are asked to choose one door. Upon choosing a door the game show host opens one of the two remaining doors, showing the goat behind it. You are now left with two doors, one with a goat and one with the car. The game show host asks you if you want to keep your door or change to the other one, what should you do?

Intuitively you might say that it does not matter if you switch doors or not, you are left with two doors therefore it is a 50/50 chance anyway. However this is wrong, it matters a great deal and you should always switch, as you will actually have a 66.66666..% chance of getting the car if you do.

One important thing to note in this problem is that it will always be a door with a goat that will be opened, never the one with the car as it would ruin the fun.
Often times explanations of the problem leave it at that, and just state this as a fact or try to go out and test it in reality with some game or simulation. These simulations or games then show this apparent paradox in action, and will have the player win around 66% of the time when switching. This might leave some with the nagging questions of "does it work with more or less than three doors?" and "why is it 66% and not 50% when the choice is between two doors".

To show this more intuitively, let's expand the problem to 100 doors instead of 3.

The 100 doors version of this problem works almost exactly like the 3 doors version. However, instead of opening 1 door at the end of the first round, the game show host will open 98 doors with goats behind them. This ultimately leaves the player with a 2 door choice at the end, like the original problem.

In the 100 door problem you can more easily see that in the first round, there is a 99% chance that a door will contain a goat, while only a 1% chance that it will contain a car. So when choosing a door in the first round, 99 times out of a 100 you will start by choosing a goat. However, now we take away 98 goat doors and leave the player with two doors, the one they chose and one they didn't.

Now you can see that initially there was a 99% chance of choosing a goat, so if the contestant stays on their door there is a 99% risk of them going home with a living lawnmower, and only a 1% chance of them going home with a fancy lawnmower. Though if they switch, these odds are reversed.

This is because of what the switching does. If you chose a goat in the first round you will always switch to the car, and if you chose the car you will always switch to the goat. Because there was such a big chance of you choosing a goat to begin with instead of the car, you will more often than not be switching from a goat to a car, and not so often from the car to a goat.

The problem actually gets more fun if more doors are left unopened, and more rounds are taken, as it really can screws with the statistics. You should try calculating the probabilities of getting a car by switching when you start with 50 doors, and take 4 rounds of choosing and removing 10 doors.

Tl:dr; Switching doors is good because you always switch from bad to good or good to bad, so when bad outweighs good to begin, with you are more likely to switch to good results than bad.

Also sorry if this was a bit rambly, it's a problem that has interested me since i heard about it and it annoys me when people discuss it without understanding what actually makes it happen. I tried coming up with a better way of explaining it, though this is the best i could manage, so please feel free to comment if you feel you have a better explanation, i would love to hear some other perspectives.