The important bit is that Monty will never choose to show you the winning door, and he will always show you a door after you pick a door and before asking you if you want to switch. He always opens a losing door.

The location of the prize and the player choice of door are independent events. Monty's choice is dependent on whether the player choice and the prize location are the same. Monty will never open the door the player picked. Monty will never reveal the prize. Those rules generate exploitable information for the player.

Here is the complete probability tree, and two possible strategies of play:

  Prize behind door A;  P = 1/3                           ALWAYS STAY strategy:
    Player picks door A;  P = 1/3 * 1/3 = 1/9               P(WIN) = 1/18 + 1/18 + 1/18 + 1/18 + 1/18 + 1/18
      Monty shows Door B;  P = 1/3 * 1/3 * 1/2 = 1/18              = 6/18
        Player stays with A;  WIN                                  = 1/3
        Player switches to C;  LOSS                         P(LOSS) = 1/9 + 1/9 + 1/9 + 1/9 + 1/9 + 1/9
      Monty shows Door C;  P = 1/3 * 1/3 * 1/2 = 1/18               = 6/9
        Player stays with A;  WIN                                   = 2/3
        Player switches to B;  LOSS
    Player picks door B;  P = 1/3 * 1/3 = 1/9             ALWAYS SWITCH strategy:
      Monty shows Door C;  P = 1/3 * 1/3 * 1/1 = 1/9        P(WIN) = 1/9 + 1/9 + 1/9 + 1/9 + 1/9 + 1/9
        Player stays with B;  LOSS                                 = 6/9
        Player switches to A;  WIN                                 = 2/3
    Player picks door C;  P = 1/3 * 1/3 = 1/9               P(LOSS) = 1/18 + 1/18 + 1/18 + 1/18 + 1/18 + 1/18
      Monty shows Door B;  P = 1/3 * 1/3 * 1/1 = 1/9                = 6/18
        Player stays with C;  LOSS                                  = 1/3
        Player switches to A;  WIN
  Prize behind door B;  P = 1/3
    Player picks door A;  P = 1/3 * 1/3 = 1/9
      Monty shows Door C;  P = 1/3 * 1/3 * 1/1 = 1/9
        Player stays with A;  LOSS
        Player switches to B;  WIN
    Player picks door B;  P = 1/3 * 1/3 = 1/9
      Monty shows Door A;  P = 1/3 * 1/3 * 1/2 = 1/18
        Player stays with B;  WIN
        Player switches to C;  LOSS
      Monty shows Door C;  P = 1/3 * 1/3 * 1/2 = 1/18
        Player stays with B;  WIN
        Player switches to A;  LOSS
    Player picks door C;  P = 1/3 * 1/3 = 1/9
      Monty shows Door A;  P = 1/3 * 1/3 * 1/1 = 1/9
        Player stays with C;  LOSS
        Player switches to B;  WIN
  Prize behind door C;  P = 1/3
    Player picks door A;  P = 1/3 * 1/3 = 1/9
      Monty shows Door B;  P = 1/3 * 1/3 * 1/1 = 1/9
        Player stays with A;  LOSS
        Player switches to C;  WIN
    Player picks door B;  P = 1/3 * 1/3 = 1/9
      Monty shows Door A;  P = 1/3 * 1/3 * 1/1 = 1/9
        Player stays with B;  LOSS
        Player switches to C;  WIN
    Player picks door C;  P = 1/3 * 1/3 = 1/9
      Monty shows Door A;  P = 1/3 * 1/3 * 1/2 = 1/18
        Player stays with C;  WIN
        Player switches to B;  LOSS
      Monty shows Door B;  P = 1/3 * 1/3 * 1/2 = 1/18
        Player stays with C;  WIN
        Player switches to A;  LOSS

If your simulation does not conform with the mathematically derived result, your simulation is incorrect.