In 1990, a reader asked the following question to the “Ask Marilyn” column in Parade magazine:
“Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?”
You initially picked number 1, and the host revealed that door number 3 is a goat. The car is either behind door number 1 or door number 2. Does it matter if you switch?
If you say no, then you agree with most of the population. However, the answer is actually yes. Switching doors will greatly increase your probability of winning the car.
This problem is a classic example of a veridical paradox, a paradox that really is not a paradox as it has a true solution. The true solution, however, is very counterintuitive and challenges even the brightest minds. According to cognitive psychologist Massimo Piattelli-Palmarini, “Even Nobel physicists systematically give the wrong answer.”
If Nobel physicists get this question wrong, how is there hope for anyone to understand this problem? The solution to this problem is actually simple, and the trouble people have with this problem is that it flies in the face of common sense. When Marilyn Vos Savant wrote the correct solution in her column, she was bombarded with angry letters claiming that she was wrong, including angry letters from many PhDs.
If you wish to understand the solution to this problem, then do not be angry or overly agitated like those professors. Understand that solving this problem requires a different way of thinking than standard probability problems. The twist is that the host reveals a door with a goat behind it, and it changes how you should approach the problem.
The chance of randomly picking the car is 1/3 at the beginning. The chance of randomly picking a goat at the beginning is 2/3. If you did not switch your choice, then you have a 1/3 chance of picking the car. This part is fairly straightforward.
However, you can double your chance of getting the car to 2/3 if you switch doors. When the host reveals a door with a goat, it changes the information you know, and as a result, changes the probability of everything. If you initially selected a door with a goat, the host reveals the other door that has a goat. As a result, the remaining door must have the car. If you switched in this scenario, you would be guaranteed to have the car.
For this scenario to work, it requires that you initially select a door with a goat behind it. The chance of initially selecting a goat is 2/3, so if you switched at the end, you would have a 2/3 chance of winning the car. The only situation where you would lose the car is if you selected the car in the first place. If you selected the car, the host revealed one of the doors with a goat, then you switched, and you would end up with a goat. Since there is a 1/3 chance of initially picking the car, there is a 1/3 chance of ending up with a goat.
Still don’t believe the solution? You can run an experiment to test this probability. If you flipped a coin a 1000 times, you would get about 500 heads and 500 tails. If you played the Monty Hall game, you would win a 1/3 of the time if you kept the choice and 2/3 of the time if you switched. How do you play? You can play a simulation here. You would have to play hundreds of times to see the probabilities play out. Luckily, the website keeps a record of everyone who has ever played the game, and you can see through the simulation records that you should switch doors.
What does the Monty Hall Problem mean? It means that common sense and intuition can lead us astray. Sometimes we have to sit down and deliberate a problem to get the right answer even if our instinct swears that it is correct. Not everything is simple and easy. If it were, the world would have no problems for us to solve.
