Let us consider situation where we have doors #1, #2 and #3. Now, these are physical doors with the number written into them, not representations in a scenario. The doors hide a car, which is randomly hidden behind one of them, two other doors have a goat hidden behind them. Therefore, each door has 1/3 probability of containing a car and 2/3 probability of containing a goat. Now, we have a scenario where a randomly player picks one of the doors. After this, the host picks another door, opens it and reveals a goat behind it.
There are two possible scenarios in which this scenario can be reached.
<b>In the first scenario</b>, the host does not know where the car is hidden and picks a door randomly. In this scenario, it is safe to say that the player always chooses door #1 and retain the same randomness due to the symmetry of the situation. It is also safe to say that host always chooses door #2 and retain the randomness of the situation. Since it was postulated that the host opens a goat door, we know that in this scenario, door #2 is a goat door, so we know by trial that the car was not hidden behind door #2.
<b>Therefore, in true random situation, we have two scenarios with equal probability to happen that describe the postulated situation:</b>
-The car is behind door #1 = You lose by changing doors
-The car is behind door #3 = You win by changing doors
This creates 50-50 odds for winning, resulting that changing doors does not matter in this situation.
<b>In the second scenario</b> the host knows behind which door the prize is hidden and intentionally chooses an empty door. Again, it is safe to say that the player chooses door #1 without disturbing the probabilities due to the symmetry of the situation. However, the door selection made by the host is not random in this scenario. He always chooses a goat door. In this situation, we do not know whether he opens door #2 or #3, so both doors remain as possible car doors.
<b>Therefore, in the situation where the host knows the location of the car, we have three scenarios with equal probability to happen that describe the postulated situation:</b>
-The car is behind door #1 = You lose by changing doors
-The car is behind door #2 = You win by changing doors
-The car is behind door #3 = You win by changing doors
I consider you smart enough to understand how the intent of the host changes the situation from this explanation. We can go through all eighteen equally likely random situations this scenario can spawn if the use of symmetry for convenience reasons hinders your understanding for some reason.