Monkey Throw Dart: Plan the Trade, Trade the Goat

I can saw a woman in two
But you won't want to look in the box when I'm through
I can make love disappear
For my next trick I'll need a volunteer

~Lyrics, For My Next Trick I'll Need A Volunteer

The Monty Hall Problem

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?

http://www.youtube.com/watch?v=vWuQVpBeqLs

The answer (and there is only one answer) is that it is to your advantage to switch your choice. A good enough explanation comes from Wikipedia, and more detailed proofs are available on the internet.

Contestants who switch have a 2/3 chance of winning the car, while contestants who stick with their initial choice have only a 1/3 chance. One way to see this is to notice that, 2/3 of the time, the initial choice of the player is a door hiding a goat. When that is the case, the host is forced to open the other goat door, and the remaining closed door hides the car. "Switching" only fails to give the car when the player picks the "right" door (the door hiding the car) to begin with. But, of course, that will only happen 1/3 of the time.

Even after an explanation, those not familiar with the laws of probability and chance sometimes don't get it. I can tell by either that glazed-eye look, or a nod of the head and a "yes" which means, " I don't understand". Some stubbornly stick to their initial (and wrong) assumption that it makes no difference if offered to switch their choice and select another door. The "if it walks like a duck and quacks like a duck, it must be a duck" logic, usually stated with conviction, does not hold water. That's unfortunate because ducks need water.

Since I believe that the best teachers are the one's who know what it's like not to understand, I usually try to re-state the Monty Hall problem using 100 doors. Behind one door is a car, behind the 99 other doors are goats. The contestant picks one, then Monty Hall opens 98 of the doors, and then asks the contestant if he would like to switch. That usually does the trick.

Of course, if that fails, just whip out the deck of cards and run through multiple Joker-Joker-Ace scenarios, and the magic of the laws of probability will appear before unbelieving eyes.

The world can be a very counter-intuitive place. Events occur that are contrary to what intuition or common sense would indicate. For trading plans, or attempting to solve real life riddles, using assumptions based on skewed data, circumstantial evidence, curve fitting, observational selection bias, over-optimization, gambler's fallacy, or emotion is no substitute for cold hard facts or solid confirmation that the laws of probability and chance will work in your favor under any condition. Anything else is just a guess. But that's okay, best guesses are necessary sometimes... as long as an open mind is close at hand.

You can "trade the plan" without all the back and forward testing if you want. It just tends to get a little expensive.