I have always been fascinated with counterintuitive puzzles. During my third year in quantitative analysis at IUPUI I was presented with the famous Monty Hall problem. It states:
Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?
Another way to say it in less words is:
In search of a new car, the player picks door 1. The game host then opens door 3 to reveal a goat and offers to let the player pick door 2 instead of door 1. Should the player switch?
Would you believe that you actually double your chances if you switch doors, even though the objects behind the doors haven’t changed? Counterintuitive indeed. You’re saying “no way, I’d have to be an idiot to believe that.”
I have never been able to explain this paradox to people without getting into a bunch of details and diagrams and examples and such, which has been pretty frustrating because your average “non-geek” person will just lose interest as soon as they see something shiny. (If detail is what you’re after, check out the all-knowing Wikipedia article on the Monty Hall problem.)
The point of this post is that I finally found a simple way of understanding this logic. Just increase the number of doors. If instead of choosing 1 out of 3 doors you chose 1 out of 100 doors with goats behind 99 of them, and the game host removed 98 doors with goats behind them leaving just your original door and one other door, switching would increase your chances from 1/100 to 99/100. This proves that switching really can increase your odds, even if nothing behind the doors is actually changed. It shows that the game host’s prior knowledge of the location of the prize plays a big part of the original odds.
If that rattled your brain, this will blow your mind:
If you were to fully develop the entire tree for all possible chess moves in a single chess game, the total number of board positions far exceeds the number of atoms in the universe.
Oh, so you disagree with that? Just like you disagreed with the Monty Hall problem? I’ll explain it later because I think I’ve crossed the threshold of geekness for one blog post. I mean, who spends their Friday posting about probabilities and statistics, honestly?
Here’s some pictures of geeks so you can at least get some pleasure out of reading this insanely boring post.