LET’S PLAY A game. Let’s say you’re a contestant on Let’s Make a Deal, Monty Hall’s game show from a generation ago. It’s your moment. There are three doors. Behind one door is a brand-new Camaro. Behind each of the other doors, there’s a goat. If you pick the right door, the Camaro is yours.
Silky-voiced Monty Hall gives you the big question:
“Do you pick door number one, door number two or door number three?”
The studio audience weighs in, and tension builds, as you think. Finally you pick, let’s say, door number one. But Carol Merrill, Monty’s assistant, doesn’t open it, not yet. Instead, Monty Hall directs Carol to open one of the other doors — door number two, say. Behind that door is … a goat. The audience titters nervously — that Camaro, still hidden, might yet be yours.
First, though, Monty is going to give you another choice. He’s going to give you both an option, and a dilemma:
“Would you like to stay with door number one,” he asks, “or would you like to switch to door number three?”
You ponder. But it’s a no-brainer, right? With two doors left, there’s a 50-50 chance the Camaro is behind either one. So you might as well stick with door number one.
Now I’m going to quote a man named Jeff Yass, from a book called The New Market Wizards. These words have made him very rich:
“The correct answer is that you should always switch to door number three. The probability that the prize is behind one of the two doors you did not pick was originally two-thirds. The fact that Monty opens one of those two doors and there is nothing behind it doesn’t change this original probability, because he will always open the wrong door.”
That’s what you didn’t consider: Monty Hall knows exactly where that Camaro is parked, and he wasn’t about to direct Carol to open a door that revealed it — no, he’s going to stretch out the game, to have fun with you, to see if, in your finger-crossed begging of the fates, you’ll switch doors.
And that’s the thing — your intuition is telling you, with two doors left, that the odds have got to be 50-50. But that’s wrong. The door you picked originally had a one-third chance of yielding that Camaro; Monty Hall picking a door he knows doesn’t hide the car won’t make it more likely that your door does.
Back to Jeff Yass:
“Therefore, if the probability of the prize being behind one of those [other] two doors was two-thirds originally, the probability of it being behind the unopened of those two doors must still be two-thirds.”
If this sort of thing makes you cross-eyed, suffice it to say that an empire has been built in Bala Cynwyd by Jeff Yass based not on numbers, but on a philosophy, a certain worldview. A world in which Monty Hall will always open the wrong door, so to speak.