TWILIGHT TRIES TO EXPLAIN THE MONTY HALL PROBLEM
“There are two doors, so it’s fifty-fifty. Even I know that,” said Rainbow Dash.
Twilight rubbed her temples with her front hooves. Somehow, her efforts to explain this to her friends were about to enter their third consecutive hour. The five mares on the other side of the table, which was completely covered with papers that displayed various diagrams, proofs, and equations Twilight had scribbled out as she tried to explain what was to her a simple if counter-intuitive math problem, regarded her with various degrees of boredom.
“It’s not though. If you switch you have a two in three chance of getting the prize you want,” said Twilight for the hundredth time.
“Not that I think that you’re wrong Twilight, because I know this is something important to you but, um, it still seems like the prize is equally likely to be behind either door. Isn’t that the way you said it was set up?” asked Fluttershy.
“Well yes, in the beginning it’s equally likely to be behind any given door, but... OK, let me start this from the top again,” said Twilight. The others groaned.
“Twilight, we’re all very hungry. We were supposed to leave for lunch some time ago. This has hardly been the ‘just five minutes’ you claimed when you began your explanation,” said Rarity.
“But it’s simple! I’m sure you’ll understand it once I explain it the right way,” said Twilight.
“Oh well,” said Rarity resigning herself to her fate, “I’m sure they’ve given away our table by now anyway.”
“So like I was saying you start with three doors,” said Twilight. Her horn glowed and three tiny illusionary doors appeared in front of her. “Now behind one door is a brand new chariot which is the prize that you want.”
“What am ah supposed to do with a new chariot? The ol’ wagon’s plenty fine for hauling apples to market. It’d just sit in the barn and rust,” said Applejack.
“You want it, OK? You just do,” snapped Twilight glaring at the pony who had interrupted her. “Now behind the other two doors are the boobie prizes. Pick one of them and you win a goat, which is bad.”
“What’s wrong with a goat?” asked Pinkie, “some of my best friends are goats!”
“It’s just...” Twilight felt her tension headache coming back. She considered just giving up, and immediately chided herself for even thinking that. If she quit now, then the last two hours would have been a total waste of time, and she refused to have thrown off her entire afternoon schedule for nothing. “That’s just the way the problem is set up,” she finished lamely.
“And how can you ‘win’ a goat anyway?” continued the relentless pony, “Isn’t that kinda like slavery?”
“It’s just a math problem,” said Twilight, desperately trying to get her explanation back on track, “It doesn’t matter what happens with the goat afterwards.”
“It probably matters to the goat,” said Fluttershy.
“Whatever! The point is, after you pick a door but before you open it the host, who knows which door has which prize, opens up a door with a goat behind it. So now there are two doors, and he asks you if you want to open the door you originally picked or switch. While you might think it wouldn’t matter, but there’s a two thirds chance of getting the chariot if you switch, but only one third if you stay with the original door,” said Twilight.
“Uh, no Twilight,” said Rainbow Dash as Twilight felt her hopes fade even further, “there’s one goat and one chariot. So it’s a one in two chance of being behind either door.”
“It isn’t though, because the host always opens up a goat door. Therefore if you picked either goat door to begin with the chariot is behind the other one,” said Twilight. Her friends still looked confused. How were they not getting this? “So since there were two goat doors to begin with there’s a two in three chance you picked one of them to start with,” she tried hopefully.
“But’cha said the host opened one of the goat doors,” said Applejack.
“Well, yes, but only after you... look, let’s do a demonstration. Then you’ll see. Then you’ll all see that I’m right,” said Twilight.
“Twilight, dear, you’re developing a bit of a, well, a twitch,” said Rarity gesturing towards her own eye.
“That’s not important! There’s understanding of obscure mathematical problems at stake!” said Twilight as she stomped a hoof for emphasis. “Now, like I said, three doors.” Three full sized doors appeared behind her in a puff of magic. “One has the chariot,” she said as the first door opened to reveal a brand new chariot.
“Wow Twilight, I didn’t know you could conjure up a brand new chariot with your magic like that,” said Pinkie.
A guilty look flashed across Twilight’s face. “Well, I didn’t so much conjure it as, er, borrowed it,” she said. “I’m sure it’s owners won’t even notice that it’s gone.”
“Twilight! You just stole a chariot?” asked Applejack with a frown.
“Only a little! Now on with the demonstration,” she said. “Now behind the other two doors are the goats.”
The other two doors opened to reveal two very confused-looking goats.
“Where am I? How did I get here?” asked the first one.
“I don’t know. I’m confused too,” replied the second. “Wait, Larry? Is that you?”
“Frank?” said Larry.
“Yeah! Wow, what a crazy coincidence running into you wherever this is. How’s the wife? Pregnancy treating her alright?” said Frank.
“Actually I was just in the hospital with her before these doors appeared. She’s kidding!”
“No way, you’re kidding!”
“No, she’s kidding,” said Larry.
“Props don’t talk!” said Twilight slamming all three doors shut. There was another poof of smoke as she scrambled the three doors. “Now Fluttershy, please pick one of the doors.”
“Um... the first one,” said Fluttershy.
“Right, so now I’ll open the door where I know one of the goats are,” said Twilight. The third door opened to reveal Frank, still somewhat dizzied from the door scrambling process. “Now, Fluttershy, decide if you want to keep that door you picked or switch. Remember you don’t want the goat.”
“Racist,” mumbled Frank.
Fluttershy thought for a moment. “Um, excuse me, Larry was it? Which door are you behind?” she asked.
“The middle one,” said Larry’s voice.
“Thank you! Oh, and congratulations on your new kid. That must be just so exciting for you,” said Fluttershy.
“Oh, I get it now,” said Rainbow Dash, “there’s a hundred percent chance of the chariot being behind the door you picked first. Got it.”
“No, there isn’t! And Fluttershy, you can’t ask the goats for the answer, it’s cheating,” said Twilight.
“Oh, I’m sorry,” said Fluttershy dropping her head and withdrawing behind her mane.
“You know what? Maybe if I change the parameters of the example this will make more sense,” said Twilight. “Let’s say that instead of three doors, there are a hundred doors.” Doors appeared all over Twilight’s library, hemming the others in. “Now, there’s still just one chariot, so ninety nine of the doors have goats behind them.” All one hundred doors opened to reveal a mob of very confused goats, who all began chattering and discussing their confusing their predicament.
“Shut up!” Twilight shouted as all one hundred door slammed shut again.
“Twilight, are you sure you’re OK?” asked Pinkie.
“Yes!” shouted Twilight, far too loudly. “I’m right, and that’s what’s important.”
“Where am I?”
“How did I get here?”
“My dream of being an unimportant prop in the display of an obscure math fact has finally been fulfilled!”
“I said shut up!” shouted Twilight over the cacophony of voices. “The important thing is the one percent chance.”
“One percent chance of what?” asked Rainbow Dash.
“Of picking the right door the first time. Of picking the door with the chariot rather than the goat,” Twilight replied.
“So, it’s not one third then?” asked Dash.
“Yes! I mean, no! I mean... in this particular instance, yes,” said Twilight already regretting confusing the issue. “I’m merely elucidating on a previously expressed point!”
There was a knock on the library's front door, and Twilight stormed over to it. “What?” she screamed as she flung it open.
She regretted in her tone a moment after she registered that her visitors were soldiers of the Royal Guard.
“Twilight Sparkle?” asked one of them. Without waiting for her confirmation he continued. “You’re under arrest for one count of Grand Theft Chariot as well as ninety-nine counts of goatnapping. Would you come with us please?”
The guards clamped their restraints onto Twilight before she could respond, and began to drag her away.
“Wait! No!” protested Twilight. “Girls,” she she called out to her friends, “remember one thing!”
“What, Twilight? We’ll remember anything you need us to,” replied Rarity.
“In a given Monty Hall situation, the odds of the desired prize being behind the door you haven’t selected is (N-1)/N, so it’s always worth it to switch! Always switch!” Twilight called out to her friends.
They were unimpressed.