The Monty Hall Problem
By: Amy Dong
Here is a famous mathematics problem: you are a contestant on a game show, and the host gives you the choice of three doors. Behind one door is a car. Behind the rest are goats. You pick a door at random, say #1. The host, who knows what is behind each door, opens one of the doors you did not pick, say #3, revealing a goat. He offers you a choice: to switch your choice to door #2, or stay with your current selection.
Being a mathematician, you know that upon choosing door #1 from three unknown doors, you had a one-third chance of revealing a car. After the host opens one of the doors, the chance that #1 contains a car stays at one-third, but the chance that #2 contains a car rises to two-thirds, since the host has revealed the goat behind door #3. So, you choose to switch.
The host opens door #2 for you. A beautiful, sleek Ferrari gleams red under the studio lights. The audience roars as you take home your prize.
...Being generally unversed in all instances of math save for the few glimpses of high school algebra that you remember from a long time ago, you know that you have two choices. There is the door that you have picked, plain as day, and the door that lies a few feet away. You decide that you would rather regret the choice that you didn’t make, rather than the one that you did. You choose to stay.
The host opens door #1 for you. There is a goat chewing grass, its hooves slightly dirty and it’s fur slightly brown. You choke down your disappointment, but it comes bubbling back anyway.
…You know that you have two choices. There is the door that you have picked and the door that lies a few feet away. The host is smirking sweetly, as if he can see the lack of cogs running through your brain. There’s a tremble to his widespread arms that you did not see when you picked door #1.
There’s something about his smile that pisses you off. You smile at him back, and choose to switch.
The host scowls as he opens door #2 for you. You cherish the distaste in his eyes more than the sports car in his door.
...Being a mathematician, you know that the chance that #1 contains a car stays at one-third, but the chance that #2 contains a car rises to two-thirds. So, you choose to switch.
The host opens door #2 for you. As you stare face to face with the goat that has been revealed, you realize that at the end of the day, probabilities mean nothing more than the floor that you stand on.
...The host opens door #2 for you. It’s a car. Sort of. The hood is sort of busted up and there’s no roof––and not in the stylistic way. When you start it up, the engine groans and there’s some sort of horrible scraping sound that vibrates beneath your feet.
The host smiles at you, next to two goats that look better cared for than your new ride. “We’ve never said that the car was the prize.”
Here is a famous mathematics problem: the host of a game show is giving out raffle tickets for a one in a million chance to compete for a prize car. You buy a ticket just for fun, and buy another few for your sister. At the end of the day, nobody wins, and no one learns anything.
Here is a famous mathematics problem: you are a contestant on a game show, and the host has just revealed an infinite amount of doors. Behind one door is a car. Behind the rest are goats. You are asked to pick a door at random. The host, who knows what is behind each door, opens every single wrong door that you had not picked, leaving only your original choice and door #33500.
You switch to #33500 immediately and score yourself a car. The host scratches his head sheepishly.
“I guess we didn’t really think that one through,” he laughs awkwardly.
Here is a famous mathematics problem: you are a contestant on a game show, and the host gives you the choice of three doors. Behind one door is a car. Behind the rest are goats. You are asked to pick a door at random.
Here is something the host did not account for. You have always been a naturally indecisive person. You ponder your options endlessly, and you doubt yourself before you can confirm your choices. You ask your friends to choose your hangout locations, and you ask the waiter to recommend your dishes. You don’t get invited to truth or dare games anymore.
“Please, make a choice. Any choice.” The host pleads with you under the studio lights. Most of the audience has left. His TV time is dwindling. The producers are shaking their heads.
You bite your nails anxiously as you weigh the consequences of door #1 to door #2 to door #3. You feel eyes all around you even though there are none.
When the host himself finally leaves, out of a job and out of a studio, you stand there alone. You and your three unopened doors.
The host shows you all three of the doors, in the end. You stand in shock as three identical cars stand, gleaming in a row.
“It’s all for the drama, anyway.” The host mutters, lighting a cigarette backstage. You stand by him, silent. “It’s the season finale. You happened to get picked. Everyone likes underdogs, right?”
You glance back beyond the curtain. The crowd is screaming for you, as people begin to clear out. There is still confetti on the ground, and the janitorial staff is having a field day.
“Oh.” Is all you say when the host hands you the paperwork. You owe him money, for the publicity, he says.
“I hope you understand.” He says, rising tiredly. “We’ve been giving out too many goats. Budget cuts, you know?” You do not. “Maybe we can get you something secondhand. How about a nice Toyota?”
Here is a famous criminal law problem: to whom do the charges for the murder of a game show host apply to? The contestant, or the goat?
The host sits down tiredly. “There is no prize,” he tells you evenly. “There never was. There is only an outcome, and it will never matter.”
“That’s too bad,” you tell him instead. The lights are down, and you can see that there never was an audience either. Just you and the stage, and some prop doors that can’t even open. The host’s makeup looks pasty and smudged without the glare of the studio lights. You suddenly feel tacky, in this suit.
You leave the studio, suit and all, and stroll down to the nearest pet store to adopt yourself a prize instead. You elect to name your outcome Cotton. Cotton nuzzles your palm affectionately.
“Would you like something to drink?” you ask instead. His posture is stilted, and even though he smiles to the audience, you know those eyes that look like a drowning man.
“Uh?” He blinks at you, startled. No one ever offers him anything, after all. Not even a goat.
“Like, tea, or something?” You gesture vaguely with your hands. He’s still standing next to the three doors, hands still outstretched, as if frozen. He really might be. “Something warm? Hot chocolate? I don’t know.”
“Are you,” the host gestures sort of tiredly to the door, “not gonna to pick something?” He’s putting on the airs of reluctance, but the shift in his brow and the slant of his mouth betray his real desperation for something other than television glamor.
“I think,” you reply softly. The microphones won’t pick this up. “You need it more than me.”
“Was there ever a choice?” whispered the game show host as the lights begin to dim. “Do we ever get a say in any matter? What do the numbers, the probabilities, the chances mean, really, when all we are going to get in the end is a yes or a no. Did you ever have a chance to win, when the world itself has been playing against you? What a foolish decision, to come on such a show. Did you know, darling? That we were already doomed from the beginning? Did you already know that it wasn’t ever the matter between the goat or the car, but the matter of whether we even got to pick one or the other?”
Here is a famous mathematics problem: you are a contestant on a game show, and the host gives you the choice of three doors. Behind one door is a car. Behind the rest are goats. You pick a door at random, say #1. The host smiles, and goes to open it.
“Aren’t you going to ask me if I want to switch my choice?” You ask, flustered. This is not how the story goes.
“Oh, darling.” The host turns to you, eyes wide and camera-ready. Their hair is slicked and plastic, and their smile is more menacing than you remember a few minutes ago. You can no longer see the edges of their facade. “You’ve made your choice, already.”
“But–” you begin to protest. You have planned out every possible outcome, until this one.
“Foolish mortal,” the host snarls under the studio lights. “Why would you ever make a decision without fully committing to it?”
Door one opens. There is a goat, but its eyes are far more predatory than you remember them to be.