The following problem was named after the American TV presenter Monty Hall who hosted the show „Let’s make a deal“.
Before the contest, three closed doors are presented to the contestant. We tell them that there is a car behind one door, and goats behind the remaining two doors. The contestant does not know which door „hides“ the car and which ones the goats, and the arrangement of cars and goats is random.
The game consisted of the contestant choosing one door. After he had made the choice, and before he had the chance to open it, the game host opens one of the remaining unselected doors behind which there is a goat and offers the contestant the opportunity to change his choice, that is, to abandon the originally selected door and choose another (not yet opened) door.
Think: Should the contestant choose another door?
This is a good moment to stop reading the text and try to put yourself in contestant’s shoes.
What would you do?
Run the simulation several times and test your hypothesis.
Solution:
The initial probability that the contestant chose the door behind which there is a car is 1/3 (because there is a car behind only one of the doors). This means that the possibility of choosing the „wrong“ door equals 2/3. Let us suppose, for example, that the contestant chose door number 1. At that moment, the probability that the car is behind the door number 1 is 1/3, while the probability that it is behind one of the remaining two doors (number 2 and 3) is 2/3. After that, the host opens one of the doors (not the one that the contestant chose) and shows that there is a goat behind them. The game host asks the contestant whether he will change his choice or not. Most of the contestants were naive and thought something along these lines: „There are two doors remaining, one of them with the car, the other one with the goat. Therefore, the probability that my original choice was correct is now 1/2. I won’t change my choice.“ However, this is a bad move because the contestant just missed the opportunity to increase his chances of winning the car. Why? Let us go back to the beginning of the story. The contestant chose door number 1 and the probability of him winning the car at that moment is 1/3. This means that the probability of the car being behind door number 2 or 3 is 2/3. The game host opened one of the doors (2 or 3) which had a goat behind them (let us say, door 3). We should now correct the conclusion from the previous sentence: The probability that the car is behind door number 1 is 1/3, and the probability that the car is behind door 2 or 3 is 2/3. However, we know that the car is not behind door 3, meaning that the probability that the car is behind door 1 is now 1/3, while the probability that it is behind door number 2 is now 2/3! Therefore, by changing his choice of doors the contestant would increase his chances of winning the car two times!
