Door Number Three

There's something non-intuitive about the Monty Hall problem. For those that don't know, it's a classic probability problem named after a game show hosted by one Monty Hall. You're faced with three doors. There's a prize behind one door, and nothing behind the other two. You get to pick a door, then Monty has an assistant open one of the doors where the prize isn't. You are then asked, "Do you want to keep your first choice, or switch to the other unopened door?"

The non-intuitive thing is that you double your odds of winning the prize if you switch doors. Why? The usual explanation maps out a bunch of conditional probabilites and is all very irrefutable, and proves that two times out of three, the prize is left behind the third door. These explanations seem to imply that somehow, after Monty does his thing, the door you chose has somehow changed. It's as though the universe spookily singles out your decision making process for ridicule or something.

That's why the equation approach somehow lacks the emotional impact that I want. I want to feel the answer. I don't want to be left feeling like somehow reality is having a prank on me.

So how about this: before anyone picks anything, let's say that the prize is a fuzzy cloud, distributed equally behind all three doors. Ok, that sounds a bit quantum mechanical, but why not?

When you pick a door, you lay claim to a 1/3-ness of the prize. That leaves a 2/3-ness of the prize behind the other two doors. Now Monty actually opens one of the doors! The state of the system goes from being totally unknown to only partially unknown. The system is still partitioned into two parts: the part you get to take home, and the part you don't. You still have your 1/3-ness of the prize, but the 2/3-ness of the prize that's not provisionally yours now has only ONE door to hide behind. In other words, there's twice as much of the prize's wave-function behind that third door.

Maybe this is old news, but I don't have time to read everything. Apologies if I've re-invented the wheel here.