The other day while browsing through the archives of Dean’s World I came across a fascinating little nugget of a post called The Brainteaser That Changed My World. I love brain teasers, and this one was especially intriguing:
You find yourself on a game show called "Let's Make A Deal." The game is very simple. There are three doors: door #1, door #2, and door #3. Behind one door is a million dollars. The other two doors contain worthless joke prizes. All you have to do is pick which door you want to open, and you get whatever is behind it. But you only get to open one door. By simple math, then, you obviously have a 1 in 3 chance of picking the correct door and becoming an instant millionaire.
You pick a door. As soon as you tell Monty (the gameshow host) what door you want to open, he stops and says, "Okay, you've made your choice. Now, I'm going to do what we always do here on this game. I'm going to open one of the other two doors for you that I know has a booby prize." And he does so. Then he asks, "Okay, now, would you like to stay with your original guess, or would you like to switch to the other door that's still closed? You only get one shot, so do you want to stay with your original choice, or switch?"
Here's the question: is there any compelling reason to switch doors?
Of course not, I thought. Why would you switch doors? They both have a one in three chance of being correct, or maybe even a one in two chance, but neither has a clear advantage over the other.
But then I had an epiphany and thought, “Yes, by all means switch doors.” Given that you initially choose one door out of three, the door you choose clearly has a one in three chance of being correct. The removed door has a zero in three chance of being correct, which means the third door (the only other door remaining) must have a two in three chance of being correct. Nothing could be more obvious, right?
I explained the problem to a few coworkers, and they all thought I was insane for insisting that it’s best to switch answers. Dance with the one that brung ya, they said, especially when there’s no reason to switch partners, which there’s definitely not. When faced with my irrefutable logic, they said I was arguing semantics, which doesn’t apply to mathematical problems.
I don’t remember the last time so many people told me I was wrong.
So I came home and shared the story with my wife. She too came to the conclusion that switching has no advantage over sticking with your first answer, but nevertheless agreed to help me do a practical test. Before we could perform an experiment, I was able to pick the brain of my father-in-law, a computer programmer who has a degree in math. I thought surely he would agree with my conclusion. But he said the same thing everyone else did: “I see what you’re saying, but I still think you’re wrong.” I should note, however, that my mother-in-law agreed with me. I love her.
Anyway, Mrs. Happy and I set up the experiment. I took three playing cards (taking the place of the three doors, which we don’t have), one of which was the ace of spades (taking the place of the million dollars, which we also don’t have), and placed them face down. She chose one. I removed one of the other two that was definitely not the ace of spades. She then changed her choice, turned over the remaining card, and marked down whether her first choice or second choice was correct. We did that 100 times. The first choice turned up the ace 35 times, and the second choice turned up the ace 65 times. Basically, the first choice has a one in three chance of being correct while switching will win you the money two out of every three times. So basically, I was right and everyone but my mother-in-law was wrong.
Here’s my point: My wife supported me and didn’t try to make me feel stupid even when she thought I was dead wrong. She always left open the possibility that I had come to the right conclusion and even took time out of her busy day to help me perform a tedious test of an inconsequential problem, because it was important to me. That, my friends, is a good wife.