You come to the final phase of a game where you can win a car. In front of you are three doors. Behind one of the doors is a car. If you open the correct door, the car is yours.

When you have indicated which door is your choice, the host of the game (who knows which door the car is behind) will open one of the other two doors and show you there is nothing behind the door he has just opened. Now you have the opportunity to switch your choice to the other door which is not opened or stick to your original decision.

Will your chance of winning the car increase by switching your choice? Or is it true that switching or not the chance of winning is still half and half?

There is a one-third chance of choosing the right door if that is the only opportunity you have. Without switching your original decision after an incorrect door is revealed and still winning the prize requires you making the correct decision in the first place. Your chance of winning the car doesn't go up to one half just because now you know one door is definitely out of consideration. (That knowledge will be to your advantage if only it exists before you make the choice.) Your chance of guessing it right is therefore one out of three.

Suppose your strategy is always switch your decision after the host reveals one of the wrong doors. (There is at least one door other than your choice which has no car behind it). Winning the car requires you choosing one of the two doors which has no car and the chance of doing it "right" is two out of three.

With no switching strategy, you need to guess the right door. Probability of winning is 1/3. Always switching your decision, you need to guess the wrong doors. Probability is 2/3. Isn't that a big deal?