The Monty Hall Paradox
Since we had a lot of fun with the last math puzzle I posted, I’ll post another one that I just ran across. It is called the “Monty Hall” paradox. It goes as follows:
You are a contestant in a game show. There are 3 doors. Behind one of them is a beautiful Ferrari motor car. Behind the other 2 are a goat. You are asked to choose one door. For the purpose of argument let’s assume that you chose Door 1. Now, let’s stipulate further that the host, Monty Hall, KNOWS which door hides the car and which ones hide the goats. Now BEFORE he opens the door you first selected (Door 1) he opens Door 3 revealing a goat. He then asks you if you would like to change your pick. Should you?
The surprising answer to this problem is YES, you SHOULD switch. If you switch to Door 2 you have a 2/3 chance of winning the car. If you don’t change your selection you have the same chance (1/3) that you had originally! Why?
It’s counterintuitive, but the problem you are faced with when one Door 3 has been opened is DIFFERENT from the problem you originally faced. There are several mathematical proofs which support this answer. The simplest is as follows:
If you DON’T switch your choice, your chances of winning the car are the same as when you started (1/3). At the same time, if you consider Doors 2 and 3 as a set, the chance that that set of 2 doors hides the car is an additive quantity (1/3+1/3=2/3). Now, when Mr. Hall opens door 3 (revealing a goat) he is actually providing you with additional information. You now know that the chance that Door 3 hides a Ferrari is 0. The probability that the set of Door 2 + Door 3 together hasn’t changed, however, and REMAINS 2/3. Since we now know that the probability of Door 3 hiding the car is 0 (because it has been opened revealing a goat) the 2/3 probability that the set originally had, MUST now reside completely in Door 2. Therefore, the probability that the Ferrari is behind Door 2 has doubled to 2/3. In essence, you have more information once Mr. Hall has opened the door. By switching you take advantage of that additional information and, therefore, improve your odds. You might still be wrong (1/3 of the time) but you would have better odds (2/3 of the time). The following diagram may make this easier to visualize:
If you didn’t get the correct answer, don’t feel bad, there have been Nobel Prize winners that have gotten it wrong. Interestingly, when this problem was used to create an experiment with pigeons and tasty seeds, the pigeons learned QUICKLY to switch!!! (Herbranson and Schroeder, 2010).
The problem was originally created by Steve Selvin in a letter to American Statistician in 1975. It was popularized by Marilyn vos Savant in 1990 in Parade Magazine (in her column, “Ask Marilyn).