OK. I too have been agonising over the Monty Hall problem.
I understand it now. Here're my notes.
Pick a wrong door: 2/3
Pick a correct door: 1/3
Pick a wrong door: 1/2
Pick a correct door: 1/2
Switching doors, is the only way to elect the odds of 2/3 and end up with the correct door.
The switch is necessary if you want to end up with the correct door, by betting that you picked a wrong door initially.
You get the 2/3 odds, because the game show host knows which door concels the car and he is not allowed to open that door. That means you have a 2/3 chance of making him tell you which door is the winning door, by constraining his choice in doors to one.
The resulting system, of two doors is not evenly distributed. That's why staying with your door leaves you with a 1/3 chance of success and switching leaves you with a 2/3 chance of success.