Monty Hall Problem

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If you are like me, you watch a lot of Mythbusters in your free time. Mythbusters takes common misconceptions or “myths” and tests them for accuracy. During one particular episode, Jamie and Adam explore the infamous “Monty Hall Problem”. I was insatiably curious after watching this episode and I wanted to find out more.

So what is the Monty Hall problem and where does it come from? The Monty Hall problem stems from the popular “Let’s Make A Deal” game show and is actually named after the game show’s host, Monty Hall. It presents the issue that people are illogical when making certain probabilistic decisions. In “Let’s Make A Deal”, Monty Hall would present three doors to the guest, two of which would have a less than desirable prize, such as goat, and one would have an expensive prize, such as a car. The host would then open one of the doors to reveal a goat, and explain to you that there is still a car and a goat behind the remaining doors. He asks you to choose between the two doors. You, as the guest, choose a random door. Monty Hall then gives you the opportunity to switch doors. You decide to stick to the door you chose because there is a 50% chance either way you would win. This would be incorrect though, for the math tells us a different story.

[youtube]https://www.youtube.com/watch?v=C4vRTzsv4os[/youtube]

This video briefly explains the math behind the Monty Hall problem. In essence, if you decide to switch, you have a two thirds chance of winning, against the a one thirds chance winning by sticking. This is because of the three options to this game.

The first choice is to switch from door 1 to door 2, which reveals a goat. You lose. The second choice is to switch from door 3 to door 2, which reveals the car. You win. And finally, you switch from door 1 to door 3, which reveals the car. You win. Adding up the probabilities, proves that you have a two thirds chance to win the car by switching doors.

If the probabilities are so simple, then why do people still fail to switch doors? In an study by Wim De Neys and Niki Verschueren, they tested to see if they could find a connection between people’s working memory capacity and their success in solving the Monty Hall problem. While they did find that people with larger working memories did do better overall in solving this dilemma, the percentage of people in any condition that solved this problem did not go over 20%. Proving that even people with large working memories fall victim to the Monty Hall problem.

This study proved that there is something else going on with this problem. In a different study by Ana M. Franco-Watkins, Peter L. Derks, and Michael R.P. Dougherty, they discovered that after repeated trials, people do eventually learn from their mistakes and chose to switch doors, but they have a hard time explaining the math behind it. One explanation given by the researchers is that there is a dual-process model between people’s judgment and people’s choice. In other words, there is a dissonance between people’s “gut” and the incorrect probabilities they come up with (such as the incorrect 50% chance of winning by sticking). With these conflicting judgments, errors in tasks such as Monty Hall problem are likely to arise.

The Monty Hall problem is an important dilemma to study for a variety of reasons. It proves that humans can be extremely irrational in both their thoughts and their behaviors and that people eventually can learn from our mistakes.