Life is a Test: A Series of Pony Logic Puzzles

by Brony_of_Brody

Nice Ponies Finish Last

Rainbow stamped at the ground impatiently. She wasn't entirely sure why she, of all ponies, was chosen for an important Royal task that involved having to sit still and be quiet for hours on end.

She had been tasked with meeting with an ambassador from the faraway city of Colarodeo, and was to escort him back home to the airport without causing too much of a fuss. Afterwards, she was to report back to Twilight, with a report of how everything went. It sounded so simple that nothing could go wrong.

The blue pegasus waited for several minutes for him to appear. Eventually, Rainbow decided to just get up and try to find the guy himself. She was warned that he was rather elderly and would need assistance, so she scanned the crowd, hoping to pick him out. It was so much easier when you can fly and get a bird's eye view of everything, all things considered.

Suddenly, Rainbow spotted a grey pony who looked like the photograph, and she hovered over to greet him. Upon meeting him face-to-face, Rainbow noted that he really looked a whole lot older than the photograph, and had a glazed-far-away look in his eye. He was wearing a small golden badge upon a red scarf tied around his neck. Recognising it as the ambassador's mark, Rainbow knew this was the pony.

She decided to play polite and talk to him. "Well...hi? I'm Rainbow Dash..."

"Ah, good morning!" the grey pony replied, in an EXTREMELY wizened old voice. "How do you do do you do do you do...I'm the ambassado-...do you do...how do you do?...I'm the ambassador of...um...do you?...the ambassador of Colarodeo...deo...deo...deo do...do you do? I'm the ambassador of Colarode...do? Er...um, what...what was I trying to say?"

"You're trying to say, 'I'm the ambassador of Colarodeo'," Rainbow ground out, already deciding she wasn't about to get along with him. He was clearly getting on in age, and she had no patience for slowpokes.

The elderly pony stared at Rainbow in what unfortunately seemed like genuine amazement. "So am I!"

"What? No, I meant-" Rainbow smacked a hoof to her forehead. "Ugh, forget it. You're Old Oak, right?"

"...Yes," the ambassador replied, after a wait that took Rainbow too unfortunately long for her tastes.

"Okay! Great! You're coming with me, to the airship," Rainbow said with a hot impatience. The sooner she dropped of the senile old bag of bones, she figured, the sooner she could get back home. She found herself resenting the behavioural standards for ponies with titles. "Hold my hoof, you'd probably just get lost or something...stupid old geezer," she muttered under her breath.

"Rainbow Dash? Is that you?" a classy voice said from behind Rainbow. rainbow turned around, and saw one of Ponyville's resident elegant musician, Octavia. The grey earth pony was carrying a small red suitcase on her back.

"Oh hey Octavia," sighed Rainbow Dash. "How's things shakin'?"

"Things are 'shaking' quite well, thank you," Octavia said. "And who is the gentlecolt with you?"

"Oh hello!" Old Oak seemed pleased to see a new face. "How do you do do you do...I'm the ambassado...do you? How do-"

"Oh sweet Celestia we're NOT doing this again. Sorry Octy, gotta fly, see ya!" And with that, Rainbow Dash dragged the rambling old pony away, towards the airship, much to Octavia's confusion.

-

Unfortunately for Rainbow, it seemed as though she was going to have to put up with her unwanted baggage for considerably longer than she would have liked. There she was with old Oak, staring at the queue going into the airship to Colarodeo, which she could see was several hundred ponies long at least.

"Aw, jeez!" Rainbow threw her hooves up in resignation. "The line's enormous! What are we gonna do?"

"Oh, don't ask me, you're the ambassador of Colarodeo," said Old Oak, staring at Rainbow, as if expecting her to use her nonexistent authority in some way.

"No, YOU ARE! Gaaaah!" Rainbow groaned, turning her attention to the line again. "Ugh, we're going to be here all day at this rate. Can't you just show your badge or something so we can-"

But in her frustration, Rainbow had forgotten that throwing both hooves in the air meant letting go of the ambassador, and by the time she turned around, she saw he was no longer anywhere to be found.

Panicking, Rainbow took to the air, and within seconds, she saw that Old Oak was somehow right at the front of the queue, showing his credentials to a green earth pony, whom Rainbow guessed was the flight attendant. Breathing a sigh of relief, she flew straight to where he was, just as he was disappearing inside the airship. However, as she tried to get in herself, the attendant whipped out a stop-sign from under her desk, and Rainbow quickly stopped herself mid-air.

"No queue jumping, miss," she said sternly.

"No no, you don't get it," Rainbow tried to explain, landing herself onto the walkway. "He's with me. I'm escorting him home and-"

"Got a pass for that?"

Rainbow paused. Nopony had given her any passes or papers to prove she was supposed to be with him: she was just instructed to meet with him and then don't lose him. "Crud."

"Thought so. Back of the line like everyone else, please."

Grumbling under her breath, Rainbow walked all the way back onto the docking point. She angrily purchased a ticket from the pony at the end of the queue, and impatiently took her place.

She hoped that Old Oak at least remembered his seat number. It was written down on his ticket, so there's no way he could possibly forget...

Unfortunately, everything that could go wrong for somepony like Rainbow Dash very often does, and Old Oak soon found himself surveying the five-hundred empty seats in front of him, asking himself which one he preferred...

Author's Note:

I hope nopony ever has this happen to them. Unless you're one of those people who decides to sit in a restaurant 5 minutes before it closes. A special place in Tartarus awaits you if so.

1. 500 seats in an airship.

2. 500 passengers in a queue, with Old Oak at the front and Rainbow Dash at the very back.

3. Forgetting the number of his seat, Old Oak sits in a random seat.

4. All other passengers in front of Rainbow Dash sit in their seat if it's available, otherwise they will sit in a random open seat. (Old Oak is too stubborn to move if asked)

PUZZLE: What are the chances of Rainbow Dash sitting in her assigned seat?

Comments

[Eclipse_Nova]

Here are the ways the seats would be filled:
1. Everypony has the right seat
2. Old Oak chose the wrong seat but not Rainbow's, while the unfortunate passenger took Old Oak's seat, allowing Rainbow to sit on her assigned seat
3. Old Oak chose the wrong seat but not Rainbow's, while the unfortunate passenger took Rainbow's seat, forcing Rainbow to sit elsewhere
4. Old Oak took Rainbow's seat.
I will go out and say Rainbow's chance to sit in her assigned seat is 50%, however I am certain that math is involved, and my answer is far cry from correct. I am too lazy at the moment

[Duckgame4life]

9212107
Forgot an option of oak get a random,that rando gets another rando and so on witch can land wth any number of ponies with the right seat

[Coyotek4]

Without looking ahead, I believe the odds will be 1 in 500.

EDIT: … and I am wrong. That does happen from time to time.

[Andy Ray]

Let us ask ourselves a question first: what are the chances of Old Oak choosing his rightful place? There are 500 seats, of them only 1 is his. 1/500, everypony gets seated as per their tickets.

Out of 499 cases Old Oak chooses the wrong place -- the pony Old Oak stole his/her seat from will select a seat randomly. 1/499, he/she chooses Old Oak's proper seat, so everypony else gets seated as per their tickets.

So Rainbow gets seated properly, if and only if everypony before her chose to sit at their places or found a place to sit, other, than Rainbow's. Just how many cases are there? Let's see:

Old Oak sat properly;
Pony 2 sat on Oak's seat;
Pony 2 sat on somepony else's seat, so either two of the 498 remaining ponies would have to seat improperly;
Pony n finds, his/her place is occupied, and chooses a seat. Out of how many places does he so? There may be 498 seats or just two. If pony n chooses to sit on Old Oak's proper seat, then the other ponies will get their seat. It takes n ponies to get involved into a "merry-go-round" and to trade seats each with other. N is not less, than two, and is not greater, than 500. As Rainbow Dash is the last pony to get aboard, she only gets to sit as per her ticket, if n is less, than 499 (because the 499-th pony may take her seat, if nopony has taken it already). Let P(N) be "the N-th pony chose Old Oak's seat":

N = 2, P(N) = 1/499 * 499/500;
N = 3, P(N) = 499/500 * 498/499 * 1/498;
N = 4, P(N) = 499/500 * 498/499 * 497/498;
...

Notice, that for each N the probability of N-th pony choosing Oak's seat equals 1/500. Assume, k ponies participate in this "merry-go-round" and P(N = k) = 1/500. Then:

P(N = k + 1) = 1/500 / (1/(500-k+1)) * (500-k)/(500-k+1) * 1/(500-k) = (500-k+1)/500 * 1/(500-k+1) = 1/500.

Therefore for every k >= 2 P(N=k) = 1/500 (note, however, that k < 500).

That'd mean, there's a 1/500 chance, that pony 499 gets Old Oak's seat, in which case Rainbow may sit as per her ticket.

(Note, that Old Oak might sit on Rainbow's seat from the very beginning, but that would mean, Rainbow would be the second pony to choose Old Oak's seat, probability of which still is 1/500).

...I feel, there's something wrong with the solution, but... we'll see.