Say you pick one of the two non-prize doors initially. The host now opens the other non-prize door, leaving only the prize door. If you switch you get the prize door.

So switching always changes the prize door to a non-prize door and a non-prize door to the prize door. You start on the prize door 1/3 of the time and a non-prize door 2/3, so you should always switch.

"If you stay with the door you picked initially you succeed if the initial door has a car, which has a chance of 1/3. If your strategy is to switch then you succeed if your initial pick is a goat, which has a chance of 2/3."

1) Your 1-in-a-trillion chance original choice just happened to be correct

2) The omniscient host picked the correct door

Clearly, you should switch.

Draw three rows of three rectangles each. Now draw a star (representing the prize) in row one, box one, another star in row two box two, and finally a star in row three box three. This depicts all possible "game states". Now we will play the game three times, but in each case you will always choose the first box in each row as your first chosen door. Now the host will open one of the other doors. In the first row, you have two choices to open, so pick one and draw a circle in it. This door is open revealing nothing. In the second and third rows, you must not open door with the prize, so you choose the empty box in each row.

Now go through and tally up how many times you would have won if you had not switched doors and stayed with your original guess (1/3). The tally how many times you would have won had you switched (2/3).

Only one of the three doors has the car. So you have a 1/3 chance of correctly picking it and 2/3 of the time you'll pick a wrong door.

Monty opens a door that he knows doesn't have the car. Since 2/3 of the time, you've picked a goat, knowing where the other goat is means switching will guarantee you get the car in those 2/3 of the trials in which you picked a goat initially.

So if you always stay with your initial pick, you keep a 1/3 success rate over n trials. By always switching, you get a 2/3 success rate over n trials.

But here's the intuition: because the host never opens a door with a car behind it, the fact that he didn't open a particular door suggests that the door in question is the winner. Compare that to the initial door, which was selected blindly.

You picked a door. It has either a goat or a car behind it, but the goat is more likely. In which case, the remaining doors have a goat and a car behind them. And the host has just kindly revealed which one has the goat.