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Morgan Lopez

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The Monty Hall Problem

The Monty Hall Problem gets its name from the TV game show, “Let’s Make A Deal,” hosted by Monty Hall1. The scenario is such: you are given the opportunity to select one closed door of three, behind one of which there is a prize. The other two doors hide “goats” (or some other such “non–prize”), or nothing at all. Once you have made your selection, Monty Hall will open one of the remaining doors, revealing that it does not contain the prize2. He then asks you if you would like to switch your selection to the other unopened door, or stay with your original choice. Here is the problem: Does it matter if you switch?

This problem is quite interesting, because the answer is felt by most people—including mathematicians—to be counter–intuitive. For most, the “solution” is immediately obvious (they believe), and that is the end of it. But it’s not. Because most of the time, this “obvious” solution is incorrect. The correct solution is quite counterintuitive. Further, I’ve found that many persons have difficulty grasping the validity of the correct solution even after several explanations. Thus, this web page.

Before I continue, you may wish to attempt to solve this problem by yourself. You’ve a good chance to do so, because you now know not to trust your instincts in this and that you should consider the problem very carefully. Try it.

The Solution

First of all, I should say that this is not a rigorous mathematical analysis of this problem. This is a pretty simple problem, and doesn’t require any advanced techniques. However, I’ve included a link to Steve Selvin’s workup of this problem. My webpage is intended to present the solution in as clear and detailed terms as I believe are necessary.

Let’s begin with a simple diagram:

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