[cestith]

Try thinking of it this way. It may or may not be any more useful. I've seen different people come to understand the problem from different examples.

    1. Observe that 3/3 = 1. Pedantic, yes, but good for frame of mind here.
    2. Pick one of three doors. (1/3 odds)
    3. Gain information that one of the three doors is a loser.
    4. Note your odds on choosing the original door correctly are still 1/3.
    5. Note that if you change doors, there are still 2/3 doors there to choose.
    5. Note you're not going to switch to the known loser door, so if you change doors you know 100% which of the other 2/3 of doors to choose.
  

The intuition usually is that you're down to two doors after the loser door is opened, but that's not the case. There are still three doors. The host has just told you that if you trade doors, you know which door to trade for. So trade for it.

Note there's a newer version of "Let's Make a Deal", hosted by Wayne Brady, but there is no option to switch after a losing door has been shown in that version.

[thehappypm]

Step 4, kind of a mystery. Why are my odds still the same even though the situation is different?

[inetknght]

You choosing a door in the first place gave you 1-in-3 chance. But your 1-in-3 choice is deducted from the 3-of-3 choices that Monty could have had, so Monty only had a 2-in-3 choice: there's a 2/3 chance that the car is in the pool of doors from which Monty could select. Monty has a 100% chance of choosing a door without a car. Therefore, you inherit the 2-in-3 chances if you change your selection.

The first door will have the car 1/3 of the time. The second door's chances had been expanded to the remaining 2/3 percent thanks to Monty always choosing the last 1/3 door which is guaranteed to not have a car.

[Swenrekcah]

Because it isn't different really. It is always a goat door that is opened, so you don't gain any information about your door by the opening of the goat door.

I'm thinking of a number between 1 and 10, guess it. If I now tell you a number I promise is not the one I was thinking and not your number, you have no more information about if you were correct.

[SamBam]

Because it really centers on the initial premise: Monty will always open a goat door after your choice, no matter what.

So, you make a totally random choice. That choice must be 1/3 right, right? Now the thing that you already knew would definitely happen happens: Monty opens a goat door. How can your odds suddenly jump to 1/2?

Are you saying every single time you play the game, you always have a 1/2 chance of getting it right first time?

[cestith]

What you knew about the door you initially picked hasn't changed at all. What you now know about the other doors has. By giving you information about 1/2 of the other 2/3 doors, Monty has given you an extra 1/3 chance if you pick among those.