This problem is based on the game show *Let's Make a Deal*, which was hosted by Monty Hall. WARNING: Solving this problem may lead to good classroom arguments.

Contestants of the show were asked to choose one of three big doors, behind which were prizes.

Let's say that behind one door is a brand new convertible, but behind the other two are goats. You pick a door. Without revealing what's behind that door, the host opens one of the other doors to show you that behind it was a goat named Steve. He asks you whether you'd like to stick with your current door or switch to the other remaining unopened door. What should you do?

Solution

This problem was made famous back in 1975 because many very smart people did not agree with the answer provided by column host Marilyn vos Savant. Head to Wikipedia to learn more. It's an interesting story.

The simplest way to look at this problem is that when you first chose a door—let's say door A, there was a 2/3 probability that the car was behind doors B or C. That probability does not change just because the host opened one of those two doors—let's say C—to reveal a goat. Choosing B or C is still your best bet, but now you have additional information about door C. That means that if you switch your door for door B, you have a 2/3 probability of getting the convertible.

This is a great problem to discuss because there will be students—just like there were many PhDs—who will argue that this answer is not correct. This can lead to diagrams, acting it out, or computer simulations.

If you like this problem, try the Two-Kids Problem.

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