(Note: this is sometimes called a ‘problem’ and sometimes called a ‘paradox’, but whatever it is, it’s fun, so I’m doing it.)
This paradox is named after the host of the game show Let’s Make a Deal, because it is based on that show’s premise. The paradox presents a fun little game show: there are three doors. Behind one is a BraAAnd NeeEw Caaaarr!!!!! And behind the others, goats. The contestant chooses a door at random, and once the choice has been made, the host, who knows what is behind every door, opens one of the two goat doors. At this point, the contestant can choose to keep the door they have, or switch. I must confess, I’ve never actually seen the real show, because I started an episode, and there was a disappointing lack of goats, and an overabundance of screaming people dressed weird.
¯\_(ツ)_/¯ If you’re into that sort of thing, I guess take a look?
Regardless, this premise opens up a very interesting statistical paradox. I’ll be honest, this one is hard to do with text alone. I’ll give it my best, but if you’re still confused in 500 words, might I suggest this numberphile video?
Okay, time to break it down. To do that, I think it’s best if you are the contestant. That’s right, you! Come on down to the stage and try to win a car!! A BraAAnd NeeEw Caaaarr!!!!!
Alright, you’re on the stage. The lights are blinding. You can hear goats bleating, reminding you of the stakes. Sweat balling up on your forehead, you select a door: door number two, to be exact. The grinning host says something witty, and the audience laughs. With a dramatic flourish, the host opens door three to reveal a goat. It’s eyes are bugged out, and it looks almost as stressed as you. No, no, you can’t let yourself get distracted by the goat. Now you know that the car is hidden behind either door two (the door you’ve chosen), or door one. Now is the time to make that decision. Do you switch, or do you stay?
At first, it seems that it shouldn’t matter. There are two options, the car was placed randomly, so it should be 50/50, right?
You, my friend, are wrong, so very wrong. But you do not know this as you stand on that stage, so you decide you might as well stick to your guns and stay. Fanfare builds up beneath you, the host opens door two, your door, and… it’s a goat.
Here’s what you missed. When all the doors were closed, as far as you knew, every door had a ⅓ chance of having the car. So, you picked door 2. As soon as you did that, you split the doors into two categories: picked, and not picked. The door you picked had a ⅓ chance, and the two doors you didn’t pick, when put together, had a ⅔ chance. Then, door 3 is opened, and you see that only one of the two ‘not picked’ doors could possibly be the right one. But- and here’s the key thing- the category ‘not picked’ still has a ⅔ chance of being right, not a ⅓ chance. Only now, that ⅔ is concentrated, one might say, behind only door number 1. So, while you’re sitting on your ⅓ chance door 2, door number 1 has effectively doubled its chances. No matter what, when this game plays out, it’s better to switch.
This has been shown experimentally by this website, the Monty Hall Page. It’s true- when people switch, they have about a ⅔ chance of winning.
So, the next time you’re faced with this weird decision, switch. Although, on second thought, if you’re the eco-friendly type, watching your carbon footprint…
You might want to stay.
(comic from the awesome xkcd.com)