Superfact 22: Suppose you’re on a game show, and you’re given the choice between three doors: Behind one door is a car; behind the other two doors there are goats. You want to pick the car. You pick a door, and the host, who knows what’s behind the three doors, opens another door revealing a goat. Now the question is, is it to your advantage to switch door choice? The answer is yes. And that is the surprising Monty Hall Problem.
It is quite common to argue that it does not matter. You don’t know what is behind the two remaining doors so it should be 50/50 right? In a test involving 228 people only 13% chose to switch. However, you should switch.
Monty Hall, the gameshow host of the Let’s Make a Deal television game show, knows where the car is, so he never chooses the door with the car. And by curating the remaining two doors for you, he raises the odds that switching is always a good bet. By switching your choice, you have a 2/3 chance of winning the car but if you stay with your original choice, you only have a 1/3 chance of winning the car.
So why is this a super-fact? First, we know it is true. It is mathematically proven and experimentally verified that switching door is the best choice. Secondly, this was widely contested and is still surprising to people. Finally, probabilistic thinking is the key to being rational and making good decisions. This fact is true, important and disputed and thus a super fact.
One way of viewing the situation is by noting that there is a 1/3 chance that the car is behind any door that the contestant picks and a 2/3 chance that the car is behind one of the other two doors.
If Monty opens one of the two doors that the contestant did not pick there is still a 1/3 probability that the car is behind the door the contestant picked and a 2/3 chance that the car is behind one of the other two doors. However, one of the doors that the contestant did not pick is now known to feature a goat. Therefore, the probability that the car is behind the other door is 2/3.
The table below is probably (no pun intended) a better way of illustrating the situation. In the table door 1 is the door designated to be the contestant’s first choice. Monty opens one of the remaining doors that has a goat behind it.
| Behind door 1 | Behind door 2 | Behind door 3 | Result if staying at door 1 | Result if switching to door offered. |
| Goat | Goat | Car | Wins goat | Wins Car |
| Goat | Car | Goat | Wins goat | Wins Car |
| Car | Goat | Goat | Wins Car | Wins goat |
There are various other ways of explaining the situation including Steven Pinker’s approach. It is easy to test this is real life and repeated experiments and simulations shown that if you switch you have a 2/3 chance of winning.
As an example of the controversy this probability puzzle caused was Marily Savant’s column in Parade Magazine. As a side note, Marilyn Vos Savant is the person who has the highest recorded intelligence quotient (IQ) as stated in the Guinness Book of Records. In response to a question regarding the Monty Hall game show problem she wrote that you should switch. She received letters from 10,000 readers disputing this, including 1,000 with PhDs. In the long run she prevailed.
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