The Monty Hall problem

The Monthy Hall Problem is a math probability puzzle.

Let's say you have 3 doors, behind one door you have a car and behind each of the 2 other doors, you have a goat. Let's call the door with the car, the winning door and the other ones the losing doors.

At first, you have to choose one door. Then, the host who knows which doors are good or bad, opens one of the loosing door you had not chosen and asks you if you want to keep your choice or switch for the remaining door. Since you only have two doors left, you think it doesn't really matter since the car can be behind any of the two other doors. The problem is: you are wrong.

At first, you choose a random door, so the probability that you have picked the winning door is 1 out of 3 ~ 33%.

Then the host opens a loosing door: but this changes nothing to your situation or information. No matter that you picked the winning door or not: he will still be able to open one wrong door. So if you stay on your original choise, you have only 33% to win the car because opening one wrong door does not gives you information.

In probability it is said that the sum of all probabilities is always 100%. So it would mean that if you choose to switch to the remaining door you'd have 2 out of 3 chances to win ~66%. But that looks strange because each remaining door seems equal.

It is true, they are equal but only after the 2nd step of removing a losing door. So if you want to benefit of a 50% chance of winning, you have to toss a coin and randomly chose without basing your judgment on your first choice.

But  why would always switching your choice gives a 66% chance?

I said that opening the door does not give you extra information regarding to your first choice but it gives you an advantage. In fact, the Monty Hall problem is equivalent to this:

You can choose a first door (stick to your first choice) or you can choose the other 2 doors and the host will get rid of one wrong door for you (you switch), so you have to 2 out of 3 doors with one joker  : 66%.

Emotionnally, people tend to keep our first choice because they see it as 50/50 choice and it causes them pain to imagine that acting could make them loose.