Monday, February 23, 2015

Update 21 and the Monty Hall Problem

Some Thoughts

Have you ever heard of what they call the Monty Hall Problem? It's sort of a riddle. It's relationship to pop-culture may be a little indirect, but I had quite a lengthy and intense conversation about it recently, so I thought I'd talk about it.

The problem goes like this: You are on a game show (hosted by Monty Hall, that's why the problem's called that) and you've been presented with three doors. Behind one door is a brand new car, behind the other two doors are goats. After you select a door, Monty opens another of the doors and reveals a goat. He then asks you if you want to change your choice to the other door, or stick with the one you picked in the first place.

Now, here's the question: assuming you want the car, does it help your chances to switch doors?

You might think not, as Monty opening a door doesn't change whether you picked the door with the car behind it or not. However, they (whoever they are) say that you're more likely to get the car if you switch. How can this be?

Well, over the course of my intense conversation I was able to come up with an explanation I'm pretty happy with. First of all, realize that Monty never opens the door you picked, and he never opens the door with the car behind it. Now, suppose you had a spy back stage who, before he got caught by the security guards, was able to see a goat behind one of the doors and get that information to you. In this case, how do you make sure you get the car?

The answer is that you pick the door you know has a goat behind it, then switch doors when Monty offers. This is because Monty will always open a door and reveal a goat, but he won't open your door, so the only door left after he's revealed a goat must have the car behind it.

Now, back to the original situation. You don't have a spy, but you do know that 2 out of 3 doors have goats behind them. This means that two of the three choices you could make will result in the same situation as when you had the spy, whereas only one choice won't. So, as they say, you're more likely to get the car by picking any door, and then switching.

One of the main, and more deeply important, points of the intense conversation was whether probability is real or a fiction. Because, you know, if you picked the car first, then the likelihood that switching doors would get you the car is zero, and you're only going to do it once. Chance has been given a lot of credit in the past two hundred years or so, and it's probably time to reevaluate that.

Progress Report

I am moving now at a break-neck pace, at least it feels that way when I leave an argument in its barest form and move on to write the next argument. I am doing this in an attempt to finish a first draft of part 1, after which I intend to distribute it to some friends and see what they think.

Part 1 has 3 sections, and I've just finished the first draft of section 2 and begun the first draft of section 3. I also read section 2 aloud to a few people, and it was amazing how much sense it seemed to make when I was reading it to someone besides myself. My audience tended to agree. Looks like I might actually end up with a book here.

© 2015 John Hiner III

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