The School of Friendship: originally a plan conceived by Princess Twilight Sparkle to spread friendship beyond Equestria, it fought through bureaucratic interference, a tiny budget and somewhat cheap hiring standards to become the best (if only because it was sort of the only) facility for relationship education the world of Equestria had ever seen.
Despite its shaky start however, it had grown to be a well-loved place for students of all species to learn about friendship. After several months at it, they had found a rhythm of sorts, and as long as the students didn't realise the teachers were just sort of making it up as they went along, whatever worked, worked. It was on one such day that Twilight, the head teacher, came to one of the classrooms with an important announcement.
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"A field trip?" asked Gallus. The young griffon looked up from preening his blue feathers momentarily,
"That's right!" Twilight smiled. "Rarity oh so generously agreed to organise one for you students! After the last one, you all agreed that the last one with Applejack and Rainbow Dash was a success...somehow..." Twilight's eye twitched slightly, for reasons that escaped the students in the classroom. "...so I thought we could organise another one!"
"Wait, I thought Pinkie Pie was organising one for today too," Smolder pointed out, raising her clawed hand.
"Well, there's that, yes," Twilight nodded. "It's to the snowy mountains of outer Yakyakistan."
"Thought so," Smolder shrugged. "That's why I packed something warm. You know, duh, still a reptile."
"So, what?" Cozy Glow raised an eyebrow. "Are half of us going on one, and the other half goes on the other?"
"Essentially. But since I know some of you already have an ideal spot in mind, so to ensure it's fair for everyone here...we're going to have it decided randomly!"
Cozy Glow hid a smirk across her face. What no-one in the class had realised however, was that she already knew this would happen. She happened to stumble upon the lesson plan as she was about to head home yesterday, and seeing that the head office door was open, what else was a curious filly to do but look around?
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Yesterday afternoon...
It didn't take long to find the lesson plan on Twilight's desk. looking it over, she saw that the field trips was going to be decided via a draw over who got to go on which trip. One to the forests in the deer's territory, and the other in the freezing mountains of Yakyakistan.
And as much as Cozy Glow was a decent skier, she decided the lure of more friends on a more close and personal orienteering trip was a better outcome. And the cold generally stank, anyway.
She looked at the plan. They would be assigned a trip based on drawing stones from one of two random jars. Whoever picked a blue stone went to freeze their flanks off in Yakyakistan. Those who got a red stone were heading to the forests. Each jar would have six stones each, one containing only red, the other only blue. Each student would be blindfolded and made to choose one of the jars, which would be hidden under a blanket. Once they chose a jar, they'd fish one of the stones out.
Fortunately Cozy Glow was blessed with some degree of cunning, and she soon realised that there was a way to cheat.
She considered her options in her head the next morning: the morning of the trip. She was pretty sure she could sweet-talk the students into letting her be the first to pick, but the main problem was the frankly unfavourable odds. After all, there were six blue stones and six red stones to pick...
She briefly excused herself for a bathroom break, and quickly made her way to Twilight's office again. When she walked into the empty room for the second time, sure enough, there were the jars under the blanket, ready for the draw.
She glanced up at the clock. Only a one minute before Twilight showed up for the test. Sweating, took the jars from under the blanket, moved the stones around to best suit her odds, and promptly dived for a good hiding place as she heard the keys jangling around at the lock outside...
Well the obvious answer is this
split 3 stones of each color into each jar, a perfect 50/50 on getting the one you want
If there is a better answer, please tell me
9184505
Uh, It was already 50:50 from the beginning. You pick one jar with all its stones the same (50% chance of picking the all red stone jar). Cozy wanted to IMPROVE those odds.
Lie and say you can't go to Yakyakistan for health reasons.
9184909
Why are you asking me.
I haven't seen it.
i haven't seen any barrels anywhere.
I would not poison a barrel.
Don't ask me any more questions.
Mm, the best I can do is approximately 73% chance of a red stone, but the odds are also extremely high that it'd tip Twilight off that something's up.
New idea
1 red stone in one jar, all other 11 in the other
50/50 to get either 100%odds or 5/11(45.45%) odds
100+45.45=145.45
145.45รท2=72.72%
Completely missed the idea of unbalanced distribution between jars
Jar A: 1 red, 0 blue
Jar B: 5 red, 6 blue
Possibility: 1*50%+45.45%*50%=72.73%
Well, if all of the stones in one jar are of the same colour, once you've selected a jar, you would be left with a 100% chance to pick a stone of a certain colour. So far the only randomiser here is the jars: there's a 50% chance to get six blue or six red stones.
Now what, if one jar contained stones of different colours? Then with a 50% chance you'd get either of the sets. Suppose now, that there's x% chance to the stones of preferred colour from one jar and y% chance -- from the other. What would be the probability of taking the stone of preferred colour (SOPC) from any jar?
Let A be "got SOPC from jar 1", B be "got SOPC from jar 2".
P(A OR B) = P(A) + P(B) - P(A AND B).
P(A) = 0,5 * x%, P(B) = 0,5 * y% (since we need to account for 0,5 chance of selecting between A and B), P(A AND B) = 0 (can't get from both jars).
P(A OR B) = 0,5 * (x% + y%).
x% / 100 * 6 + y% / 100 * 6 = 6 (probability * num. stones in one jar = num. SOPC in one jar; in both jars num. SOPC should be 6).
(x% + y%) * 0,06 = 6, -->:
x% + y% = 100%.
Therefore, however the stones are laid, the probability of picking from either jar a SOPC is still 50%. However! When it comes to just one of the jars, the probability to get a SOPC is not either 100% or 0%, it's x% or y%! All that's needed is to take some SOPC from one jar and put them into the other. Given the number of stones in each jar, let's take 3 SOPC and put them into the other jar, making room for them. This should lead to each jar containing 3 red and 3 blue stones.
Now comes the trick! It is only logical to assume, one will take the most upper stone, no matter which jar. If in each jar the SOPC will lie on top of their counterparts, then the probability to pick a SOPC shall increase, since they are more likely to hit the taker's hoof/whatever! Again, no matter which jar!
Therefore in each jar there are 3 stones of unpreferred colour (blue) and 3 stones of preferred colour (red) on top of them. Whoever goes first, gets the red one.
...Yes?