The game's concept is that there is money placed randomly behind one of the three doors. The host has already opened one door which is empty. Now the choice is between the door that has been chosen and the other. Say the chosen door is Door number 3. Door number 2 has already been opened. So now the choice is between Door 1 and Door 3. The contestant has been given the choice of swapping his initial choice. That is Door number 3 with Door number 1. The conservative choice is sticking to the initial door but psychology says that conservative choice seeks less financial reward. Swapping the doors will fail only if the initial choice that is Door 3 had contained the money. But the probability of picking the money in the beginning itself is only 33.33%, one out of three.
Further, let us look into the three possibilities of switching. In the first case let Door 1 contain the money, hence doors 2 and 3 are empty. The initial choice was Door 3. By sticking to the initial choice, the contestant loses, while by swapping he gets the money. In the second case, let Door 2 hide the money. If the host opens Door 1, the contestant is left with Door 3, his choice, and Door 2. Here also by swapping he can win while if he chooses to stick with door 3, he loses. In the third case, let Door 3 contain the money. This is the only situation where the contestant can win if he sticks with his initial choice of Door 3. In two of the three cases, the probability of winning is higher if the contestant swaps. Thus switching the doors has a winning probability of 66.66%.
Initially, the probability of the money being behind the three doors is equal. Most of the people assume that after the host opens the empty door, the remaining two closed doors have equal probability and switching would not make any difference. Hence, the usual behavior is to stay with the initial choice. However, the contestant must switch because it doubles the probability of winning.