All right statisticians, the gloves are off. Get ready for a fight.
Home from college, my son presented me with the Monty Hall paradox (which I had encountered before with similar incredulity). With the self-assurance unique to denizens of the ivory tower, he argued passionately against my insistence that the universally accepted conclusion is a statistical fiction that has no basis in reality.
For the uninitiated, the famous problem goes like this:
You are a contestant on Let’s Make a Deal, and Monty Hall (the original show-host) offers you a choice of three doors. You choose Door Number 2. Obviously, your odds of winning the Ferrari are three-to-one against.
Monty then reveals that behind Door Number 3 is a goat. Not only are you still in the running, but your odds have just shortened to even-money.
So here’s the question: Given the option, should you stay with your original choice of Door Number 2 or switch your bet and take Door Number 1?
Most of us would say that it doesn’t matter. With two possibilities, your chances are 50-50, no matter which door you choose. So why switch?
But that’s not what Statisticians say. Rather, since your original choice left you with a ⅔ chance of losing, one of the two ways you could have lost is now removed. Consequently, Door Number 1 now absorbs the ⅓ probability that previously resided with Door Number 3. In other words, the chance of the Ferrari appearing behind Door Number 2 remains at ⅓ while the chance of it appearing behind Door Number 1 doubles to ⅔.
Mathematically, this makes perfect sense. Practically speaking, it is utter nonsense. I’m still left with two unknowns, which are just as unknown as they were before the cranberry sauce appeared. Two chances: even-money; 50-50. That’s all there is to it.
No! Scream the statisticians. We’ve proven it mathematically. We’ve even tested it, and it works.
Well, maybe they have. I don’t know; I wasn’t there. But the popular illusionists Siegfried and Roy demonstrated a lot of interesting phenomena, too, so forgive me if a remain a skeptic.
You won’t forgive me, Mr. Statistician? Okay, I’ll prove I’m right.