It's blowing my mind!

I just read a fascinating book called "The Drunkard's Walk" by Leonard Mlodinow. It's about randomness and chance and how our brains are not wired correctly to comprehend chance. It was on the "new book" shelf at the library and I thought, "It might be boring, but I'll give it a try."

It was not boring. I learned a lot, and really had my eyes opened to my lack of understanding. One example he cited has really stuck with me. It's blowing my mind!

It's called the "let's make a deal" problem. Bear with me: I realize that this might be a bit dry, but I'll try to keep it interesting.

Monty Hall had a game show called Let's Make a Deal. One of the games involved him showing three doors, or curtains, or boxes. Behind one of the doors was a fabulous prize! Like a moped, or a vacation for 2 to Des Moines, Iowa, or something. The other two doors hid goats, or something of equal un-value.

Let's pretend that you are the contestant. You must pick one of the three doors. You have a 1 in 3 chance of winning. So you pick one.

Monte, who knows what's behind all the doors, opens one of the two doors you didn't pick, and he always picks one that is an anti-prize; a goat door. So now there are only two doors remaining. You are given a choice: you may either stick with your original choice of door, or switch to the other unopened door.

Well, there are only two doors left, so its a 1 in 2 chance of winning, right? So either door has an equal chance of sending you and a friend on that all expense paid trip to Iowa.




I'd like you to stop reading for a moment and think about this. Is there any advantage to switching to the other door? Close your eyes and hum some classical music so you'll be smarter. When you're sure come back.

I could not think of any reason to switch. I racked my brain, but it always came back to a 50/50 chance. However, during the run of the show, and in computer simulations it can be shown that you're twice as likely to win if you switch to the other door.

No way! Two doors, one prize. It has to be 50%.

This is what blows my mind.

It comes down to breaking the random chance with an all-knowing observer. OK, you started with a 33.3% chance of being right, and a 66.7% chance of being wrong. The observer, who knows what is behind all the doors, opens one of the bad doors. He doesn't select a random door, he purposefully selects one of the two bad doors.

What this means is that you still have a 33.3% chance that you picked the wrong door, and a 66.7% chance that one of the other two doors is right. You're not going to pick the door that Monte showed you, because you now know it's wrong. In effect, the other door has absorbed all of the % chance of both doors. If you switch to the other door, you now have a 66.7% chance of having the right door.

It really works that way, here's a link to a page where you can try it. Just make sure you try each case enough times to be valid. (no, once is not enough. Try 30 or 40)

Does that hurt your brain? I can explain it, but do I truly understand it?