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.
I've always loved that little math tidbit... Well, "always" since I first heard about it.
Twilight shouldn't worry though... I'm betting the prosecutor is willing to make a deal.
On the next episode, Twilight tries to explain why, if two equally scrumptious piles of oats are placed equal distances apart from Cranky Doodle, the stubborn mule will starve to death before choosing either pile.
"“My dream of being an unimportant prop in the display of an obscure math fact has finally been fulfilled!” nice
oh and an explination for peoples! 3 doors, each door has a 1/3 chance of being right. The host opens one of the doors, so it ISN'T a 50/50, because you know one of the doors is wrong, meaning you are more likely to switch to a wrong one.
This is brilliant.
"Behind one of these doors, you will find freedom, behind the others jailtime for your crimes of goatnapping and chariot stealing! Now... knowing that there is jailtime behind this door... do you want to switch?"
2613384
It was inspired by a real world conversation.
And yes. I'd switch.
Broken... I am broken.
2612739
I might not be following what you're saying, but the point of the math example is that you're more likely to switch to the *correct* door if you change your choice in the end.
Basically, the odds are you chose the wrong door with your first pick (a 2 out of 3 chance of this.). When the host then removes the other wrong door, the odds are that the remaining one is the winner.
She probably should have used a chart instead of a live demonstration
Mathematician by education here. Here's why Twilight's math is right. Consider the moment right before the announcer opens a door, when all the random factors have been decided but you don't know the outcome yet. This is a complete list of the possibilities of chariot placement and door choice.
The chariot is behind door #1:
IF YOU PICKED DOOR #1: The announcer opens a goat door. Switching loses.
IF YOU PICKED DOOR #2: Then the announcer has to open door #3. Switching wins.
IF YOU PICKED DOOR #3: Then the announcer has to open door #2. Switching wins.
The chariot is behind door #2:
IF YOU PICKED DOOR #1: Then the announcer has to open door #3. Switching wins.
IF YOU PICKED DOOR #2: The announcer opens a goat door. Switching loses.
IF YOU PICKED DOOR #3: Then the announcer has to open door #1. Switching wins.
The chariot is behind door #3:
IF YOU PICKED DOOR #1: Then the announcer has to open door #2. Switching wins.
IF YOU PICKED DOOR #2: Then the announcer has to open door #1. Switching wins.
IF YOU PICKED DOOR #3: The announcer opens a goat door. Switching loses.
Out of the nine possible cases, switching wins six times = 2/3.
Caveat: This assumes that the announcer always opens a goat door and then offers a switch. If it's rigged so you're only given the chance to switch when you were correct to begin with, then switching always loses. So the real-world application of this involves a healthy dose of psychology and/or cynicism.
2613384
Even better, you can get Larry to subtly point to one of the doors from the witness stand, and work in a "Lady or the Tiger?" homage too.
And this is why, in Equestria, the Monty Hall problem probably involves chickens instead of goats.
2614251
I'll need to rewatch that episode of mythbusters....
jk jk I'm too lazy *table flip*
that was much easier...
It's a shame that Twilight never got to go through the hundred-door version, that just makes it so obvious. You pick a door, then Monty opens 98 doors and tells you that the prize is behind either the door you picked, or Door #57.
2612581 Heh, punny.
In the next episode, can she explain why 0.999... = 1, please?
but what if you picked the door that the host then opened?
2688679
because calculators are lazy and stupid
3045270
Part of the setup is that the host always picks a door with a goat, and never selects the door the contestant did.
3045275
Actually, it's not just a calculator thing:
X = 0.999... (infinitely repeating)
Therefore:
10X = 9.999...
Subtract X from both sides:
9X = 9
And finally divide to find that X=1
Took me a few minutes to really get this puzzle but now that I do I like it. Thanks.
Nice lateral thinking Flutters!
I believe that the best way to make other people understand this problem is this.
Ok, three doors, one big prize, yadda yadda. What would you choose: Open one door and get whatever is behind it, or open Two doors and get the best result?
Two doors, right? Well, that is equivalent to the deal the host is giving to you. The host just filters the worst result before giving you the choice to make the choice less evident.
I love this problem. My dad took an hour explaining it to me. At least the goat/chariot thing didn't bother me too much. Actually this reminds me of that Cat problem. I can't remember what it's called, but you put dynamite and a cat in a box and then go into an observation room. You wait five minutes and just before you go back in the room both the dynamite exploding killing the cat and the cat surviving options have happened. It's only when you enter the room that one option is chosen, but only because you observed it! So yeah.
3632123
Sounds like Schrodinger's Cat
It's frustrating when your brain does not conform to mathematical reality, isn't it?
Never mind THAT problem, there's a much more important puzzle at hand: how did I read the first chapter of this and not favourite it and miss all the other chapters?
Also
Best line.
The perils of math...
--
Small typo:
its
Also in chapter #1
its
As a huge math nerd (nerd pony is best pony) I am happy to say I learned something today. Eakin I love you more and more.
Huh, I was just thinking about this story and decided to look it up, and reading the comments made me think...
2622804
I really like the large N version. That one remaining other door looks awfully suspicious. But there's a hole that's been niggling me for a long time: why does it generalize in our favor? In other words, why do we get to assume that the three-door version has the host opening (N - 2) doors, rather than 1 door? Isn't it just as consistent with the original problem to pick one of 100 doors, have the host open only one other door, and have to decide between keeping your first choice or randomly choosing between 98 other doors? It's not nearly so obvious that switching is the best strategy in that scenario -- all 98 doors can't be that suspicious.
I'm way too lazy to write a general proof that switching remains the dominant strategy when the host opens any number A doors such that 1 <= A <= N - 2, but I think I can manage a chart like 2614679's for N = 4 and A = 1. I took a lot of shortcuts and it requires me to do some weird stuff with expected returns, though, so hopefully Horizon will still give a critical eye to my math.
Say the chariot is behind Door #1: If you pick Door #1, the host opens (say) Door #2. (It could be any door, but the effect is the same and this is easier to write). Staying wins every time. Switching (to a random choice of #3 or #4) loses every time. If you pick Door #2, the host opens Door #3, leaving #1 (winner) and #4. Staying loses every time. Switching wins 1/2 the time. If you pick Door #3, the host opens Door #4. Staying loses every time. Switching wins 1/2 the time. If you pick Door #4, the host opens Door #2. Staying loses every time. Switching wins 1/2 the time.
I'm able to convince myself that the expected values are the same if you change the location of the chariot or have the host open other doors; hopefully that's clear enough that I don't need to prove it or write it out, but see "way too lazy" above. So staying wins all of 1/4 of situations, or 1/4 overall, and switching wins 1/2 of 3/4 of situations, or 3/8 overall. 3/8 > 1/4, so switching remains the dominant strategy. My strong guess is that the 98 remaining doors in the N = 100, A = 2 game "look" something like 1/98 as "suspicious" as the one very suspicious remaining door in the N = 100, A = 98 game (in other words, the advantage in expected value of switching over staying is inversely proportional to A but always positive), and that switching is the dominant strategy for any N and A.
6552076
One-sentence version of why switching remains the dominant strategy even if Monty opens only one door out of many (again, assuming the opened door and the choice are offered unconditionally):
You have a better chance of successfully picking 1 winner from among 99 doors than you do of picking 1 winner from among 100 doors.
You know, I didn't understand the Monty Hall problem until I read this! Thanks!
i actually read a book where the famous Marilyn Vos Savant explained this problem...several times. the fact that the host is DELIBERATELY picking a "goat" door to open changes the odds,
And that's why you should never choose the 101st door in a Twilight Sparkle/Monty Hall problem.
you do you, bro