Imagine he picks one of the following ways: 1) picks randomly and opens the door, or 2) picks randomly and then, if that door has a car, switches. His

Imagine he picks one of the following ways: 1) picks randomly and opens the door, or 2) picks randomly and then, if that door has a car, switches. His application of knowledge is converting those cases that would have been thrown away in case 1 into victories for "switch". The counts bear this out. If you simulate it, 1/3 of all games go to 'stick' regardless. Either 1/3 or 2/3 go to 'switch', and either 1/3 or 0/3 (respectively) end up with Monty revealing the car.