In a rare coincidence, I spent a solid two hours considering the Monty Hall problem the other night, and today I was linked to a rather absurd discussion on the very same topic. In this post I plan to present not a new solution to the problem, but rather a new explanation that makes the most intuitive sense to me.
Here’s a quick review: the Monty Hall problem is a probability problem based on a game show. There are three closed doors. Behind one is a prize (like a car), and behind the other two are booby prizes (like goats or pigeons or something). The host asks you to choose a door—A, B, or C. After you’ve chosen a door, the host opens a different door and reveals a booby prize. You are then asked whether you want to change your mind or not. What should you do?
The answer, it would appear, is that it doesn’t matter whether you change your mind or not. However, after careful consideration, you can see that you’re better off changing your mind (!)
What does that mean? It means that if you choose door B, and the host opens door A, then you should choose door C. The discussion I linked to above and the Wikipedia provide some useful and interesting explanations. What follows is the explanation I thought up.
My Explanation
My thought process stems from one question: how is it that “changing your mind” can affect the situation? Consider what happens if you choose door A and the host opens door B. There are exactly TWO ways to arrive at this situation:
- You have chosen incorrectly (the car is behind door C)
- You have chosen correctly (the car is behind door A)
A common (and correct) explanation is that scenario 1 is indeed more likely than scenario 2. But also consider this: if the car is behind door A and you choose door A, then the host can open either door B or door C. If you did a statistical analysis—for example, have 300 contestants choose door A—you would find, on average, ~33% would have chosen correctly (because the car has equal probability of being behind A, B, and C), and ~66% would have chosen incorrectly. Suppose the correct door is door A.
Those ~33% who chose correctly, only half of them, or ~16.5% of the who group, would see door B open as in the situation I described. The other 16.5% would see door C open. For the 33% who chose door C, ALL of them will see door B open (because door A is correct). The 33% who chose door B would NOT see door B open, and would those find themselves in a different situation altogether.
Adding, we find that 16.5% + 33% = 49.5% (actually it’s 50% without rounding) of the entire sample will see door B open if door A is correct–as expected. However, of those 50%, 33% chose incorrectly, and only 16.5% chose correctly. Therefore, 2/3 of the time, it’s better to switch.
You can now easily take this explanation and let door B or door C be correct, and you will find the same result.
The Meta-Explanation
What I like about my explanation is that it can answer the question I first posed: how can changing your mind affect the situation in such a way? The answer: it can’t. My explanation describes how there are only so many ways to arrive at a particular “situation,” and each situation is reached more frequently by people who have chosen incorrectly.
If you examine my argument closely, you’ll find it isn’t much more than an expansion of the standard “you’re more likely to choose wrong” argument, but I like it because it helps me understand the solution more intuitively.
Posted by Jay 
Posted by Jay 
Posted by Jay 