Rebecca Goldin Ph.D
There’s no way of better inspiring people to think about math than to give them a conundrum where the stakes are high. So it is in the Monty Hall Game, which became famous on the program Let’s Make a Deal in the 1980s. The game gets played as follows. There are three doors, numbered 1, 2, and 3. Behind one of them is a car, and behind the other two are goats. First, a player chooses one of three doors, say Door Number 1. The host then reveals one of the other two with a goat behind it, say Door Number 2. Then the player is offered to switch to Door Number 3 – should she switch?
Our intuition may not be our best friend here. Your gut may tell you that by opening a door that has no goat behind it, you have gained no information (you already knew that there was at least one goat behind Door 2 or Door 3). So your chance of winning is just 50/50. But here’s where it gets confusing. Your best strategy depends on how the game really played.
If the rules require the host to open one of the remaining doors with a goat behind it, then you should switch. When you picked Door Number 1, there was a 1/3 chance that your door has the car, and 2/3 chance that the car is behind one of the other two doors. After the host reveals that one of them has a goat, if you switch to the remaining door, you’ll have a 2/3 chance of winning.
On the other hand, if the host is not required to offer a switch, the odds may not be on your side to switch. And depending on what rule he’s following, your chance of winning or not could rise or fall considerably.
Take for example, the case where the host knows there’s a goat behind Door Number 2, and no matter what choice you make, he intends to open that door. So if you choose Door Number 2 right at the beginning, you lose. If you don’t choose it, then he opens Door Number 2 and offers you the chance to switch. Your chance of winning is now 1/2, regardless of whether you switch or not.
On the other hand, the host could be truly mean-spirited. If you choose a wrong door, he plans to open it and show you that you got yourself a goat. If you choose a correct door, you are offered to switch (and even offered money to take it). Now the chance of you winning if you switch has been reduced to zero!
This problem illustrates the importance in mathematics of having a very clearly stated problem. The problem of whether you should switch is not answerable without a clear statement of the host’s strategy.
Recently, the mathematics of this problem has been applied to psychological experiments involving choice rationalization. The New York Times’ John Tierney has done an outstanding job at explaining the mathematical mistake made by some researchers in establishing a baseline for preference while conducting these experiments. The graphics are extremely illustrative, both in establishing how the Monty Hall problem is related, and in how an excellent article about a confusing yet inspiring math brain teaser can be written.
April 22, 2008 at 9:24 am |
Consider this variant: the host MUST open one of the two doors not selected by the player, but the host DOES NOT KNOW where the car is, so the host chooses randomly and HAPPENS to open a door to reveal a goat. Now, should the player switch?