What is the Monty Hall problem?

The Monty Hall problem is a probability brain teaser that many consider to be counterintuitive. The problem is stated as follows: Imagine you are a contestant on a game show and the host shows you 3 doors. Behind one of the doors is a car and behind the other two are goats. The prizes have been randomly assigned to the doors and you want to win the car. The host allows you to pick one of the doors as a first selection. Then, the host opens one of the remaining doors and reveals a goat is behind it. The host now allows you to decide if you want to switch from your selected door to the remaining unopened door. Should you switch, to maximize your probability of winning the car?

Most people believe switching does not improve your probability of winning the car. However switching does increase your probability of winning the car from 1/3 to 2/3. An intuitive way to understand this problem is by first considering the initial state. At your first selection the probability that you have selected the right door is 1/3 and the probability that the car remains behind one of the two unselected doors is 2/3. Then the host reveals more information. The host reveals that one of the unselected doors does not have the car behind it. The probability that the originally unselected doors must have the car behind one of them is still 2/3, but you now know one of them specifically which does not have the car. Therefore, the remaining unopened unselected door must have a probability of 2/3 to contain the car, and your originally selected door must have a probability of 1/3. When offered the chance, you should switch to maximize your chances of winning the car.