(Warning: this one involves probability, so if you don’t understand probability you may not follow it. I’ll have to do one on probability at some point.)
It’s called the Monty Haul Problem, but the heck with that. Let’s tell a story.
A travelling salesman stops for the night at a farm, and the farmer informs him he’ll be sleeping in the barn. “There are three doors in the barn,” says the farmer. “Behind one is the comfiest bed this side of the Ozarks, but behind the other two are heaps of dung. Now, I’ll let you pick a door, and then I’ll tell you one of the doors you didn’t pick has dung behind it, and if you want to switch to another door, you just say the word. But once you’ve decided on a door, that’s where you’re sleeping.”
The salesman picks door number one. “Well, door number three has dung,” says the farmer with a grin. “Now, you sure you want door number one.
What should the salesman say here?
Before you break something raising your hand, I’ll tell you: he should switch to door number two, and not because I know that that’s the one with the comfy bed. I don’t know; I’m not the farmer. We’ll ask him in a bit.
Turns out, if someone gives you a choice between two hidden bad options and one hidden good one, then takes away one of the bad ones and asks if you want to switch, you should. It’s not a guarantee, but it’s a question of odds. And no one believes this.
Let me try to explain.
You have a 1/3 chance of picking correctly, right? If you get to pick and then let it ride, you still have a 1/3 chance of being right, right? Wrong.
Let’s examine: by removing one bad option, the farmer has made it so that if you switch, the only way you lose is if you picked right in the first place, right? If you pick right, then switch away, you’re guaranteed of a loss here and winding up sleeping in dung, but if you pick wrong, because there’s now only one other option, the right one, you will be sleeping pretty.
So accepting that the farmer has changed the rules of the game from “randomly pick the correct door” to “randomly pick the incorrect door, then switch to the guaranteed correct door,” you can now hopefully see that in the first choice, your chances of picking the incorrect door are 2/3. Now I will take 2/3 over 1/3 any day of the week.
But let’s frame it another way, if you’re still having trouble.
Suppose, instead of one salesman, we have three, one for each door. None of them know anything more than the first, and they all get the same spiel. And since I’ve asked the farmer, the bed is behind door number one. As we’ve seen, salesman 1 picks door one, but then because he was paying attention, he changes to door two. Dung for him. This isn’t shaping up well.
Salesman 2 picks door two. The farmer shows that door three is dung as before. Salesman 2 changes to door one. Winner winner chicken dinner.
Salesman 3 picks door three. The farmer shows that door two is dung. Salesman 3 changes to door one. Also sleeping pretty (not together; these are on successive nights).
Two of the three salesmen won by changing; one lost. If they’d all stayed put on their choices, two would have lost and one would have won. If you take this and extrapolate out, we get that 2/3 : 1/3 division again. I could do a random distribution of beds, picks, and removals (bearing in mind that the removal must be a dung door) and get the same breakdown, give or take a few decimal places.
Counterintuitive, ain’t it? Don’t feel bad; some terribly smart people have agreed with you.
