Audette-O-Blog

I finally figured it out! Yippee! I’m so excited! …yeah, I know. I need to get out more.

Here’s my problem – I’m pretty smart. BUT… I think in a very odd sort of way. I think in very non-linear patterns. My thought processes are abstract and amorphous. I don’t think like a Mathematician thinks. I think like… well… an Artist. So, while I have the ability to understand complex problems that would completely elude a lot of people, it sometimes takes me a while to find an approach to a problem that makes sense to my brain.

For quite a while I’ve struggled with the so-called “Monty Hall Problem.” And, I was fairly convinced that it was bunk. I was almost to the point of believing that all of those Mathematicians were full of it, and one simply did not increases one’s chances of winning by switching their initial guess. That’s what I was beginning to believe… until today. I finally happened upon an approach that makes sense to my brain. And, I can now assure you that you do indeed increase your chances of winning by making the switch. And, the solution seems ridiculously simple to me now. I don’t understand how I didn’t see it before.

If you’re not familiar with the problem, here is the explanation from Wikipedia.

And, after much thought, here’s what I came up with:

At the beginning of the game, before any doors are opened, your chances of picking the door with the car behind it is 1 in 3. Your chances of picking a door with a goat behind it is 2 in 3. So, the odds that you’ve picked a goat is twice as likely as the odds that you’ve picked the car. I’m sure everybody can agree with, and easily digest that fact.

So, you pick any door… let’s say you pick door number one.

Now, Monty Hall opens a door to reveal a goat. It could be door two or three, it doesn’t matter, but lets say he opened door number three. You’ve picked door number one and Monty Hall opens door three to reveal a goat. Now Monty asks you if you’d like to stick with your original choice, or if you’d like to switch to door number two. Indeed, switching will in fact raise your likelihood of winning the car. Here’s why:

At the outset, you picked door number one. And, there’s a one in three chance that you picked the door with the car behind it. There’s a two in three chance that you picked a door with a goat. Monty opens a door to reveal a goat, but that doesn’t do anything to change the one in three chance that you picked the correct door. So, the odds are still one in three that you were correct on your guess, and they’re two in three that you were incorrect. So, by switching your choice, you flip the odds. Does it make sense? You were PROBABLY wrong with your original choice. Right? Now that there are only two possibilities left, and you were PROBABLY wrong with your original choice, if you change your original choice, you are now PROBABLY correct.

I don’t know if that makes sense to the way your brain works, but it makes perfect sense to my brain. I guess I owe some Mathematicians an apology.