1. Probabilities. The probability of something happening is dependent on constraints. Given a ‘free' - unconstrained - choice between two actions, there is a probability of one half of either one being chosen. These probabilities may change if some constraint is put on the choice.

2. In the Monty Hall problem, the contestant chooses on two occasions. In the first, unconstrained, he stands a one third probability of choosing the door hiding the car, two thirds of not doing so. In the second, unconstrained, he has a probability of one half of picking the car, one of the two other options having been eliminated.

3. If the contestant constrains his second choice, this last probability may change. If he decides always to stick to his first choice, the second ‘choice' is now forced. He has essentially eliminated the probability of one half of winning on the second choice; he wins only if his first choice was correct Thus the probability of winning overall remains one third.
If he decides always to switch, then he has anulled the one third probability of being right first time. He wins if his first choice was wrong with a probability of two thirds, since Monty has eliminated the other possible wrong first choice. If his first choice was right - with a probability of one third - he has given it up, and loses. (In this case, it might appear that he can lose in two ways, depending on Monty's choice of door, but neither of these possible events alter the contestant's choices, and hence lie within the one in three chance of the his first choice being wrong.)

4. If the contestant decides before his first choice to switch at his second chance, he converts his two chances of choosing wrongly to winning choices, since Monty has to eliminate the other wrong choice. He does this at the expense of losing his one in three chances of having been right first time.

5. So decide to switch every time, and never change your mind at the second choice. If you do, half the time you'll be wrong.