Saturday, November 23, 2013

The History of the Monty Hall Problem

One of my very first blog posts was about the Monty Hall Problem. This is an extremely classic example of a probability paradox. Let me quickly describe the problem:
Pretend you are on a game show, and the host gives you three doors to select from. One of these doors has a car behind it, while the other two have goats. Let's say you select door number one. Then, the host (who knows where the car is) opens another door to reveal a goat. Let's say he opens door number three. You are then given the option to either stick to door one or switch to door two. Does either strategy have an advantage?

The common answer would be that it is 50-50, and there is no advantage either way. However, the correct answer is that there is only a 1/3 chance of winning by staying put, and a 2/3 chance of winning by switching. Click here to learn why.

This problem was first posed by Steve Selvin, but it was popularized by Marilyn vos Savant in 1990. Vos Savant is famous for once having the highest IQ in the world, as well as her "Ask Marilyn" column in Parade Magazine.

One week, her column was about the Monty Hall Problem. She posed the question, and then explained her reasoning as to why there is a 2:1 advantage for switching. This created a pandemonium of angry readers who insisted that she was incorrect, and furthermore, accused her of adding to the problem of innumeracy and lack of mathematical intuition in America. Some of these complaints came from a statistician at the National Institutes of Health, the deputy director of the Center for Defense Information, and professors at George Mason University, University of Florida, University of Michigan, Millikin University, Georgetown University, Dickinson State University, Western State College, and more. Even the legendary Paul Erdős couldn't wrap his brain around the paradox.

This problem has continued to baffle everyone it encounters, from average people to accomplished mathematicians. In 2010, Walter Herbranson and Julia Schroeder of Whitman College performed an experiment to see if playing the game multiple times could end up refining the player's strategy. The human test subjects failed to revert to the optimal strategy and switch doors in the experiment. However, when the test was performed on pigeons, with mixed grain as the prize, they were able to pick up on the fact that switching doors gave them the best chance of success. The fact that a pigeon can do better than a human in this situation is fascinating to me.

The Monty Hall Paradox is something that reminds us of how humans are not wired to understand probability and statistics. This is why people can be fooled by mathematical scams and why casinos are packed full of gamblers. If our math curricula put an equal focus on probability and statistics as it did on algebra and calculus, then our world would have much better math minds and critical thinkers in general.

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