A $20,000 scholarship went to the winner (2017 MathCounts Final)

"In a barn, 100 chicks sit peacefully in a circle. Suddenly, each chick randomly pecks the chick immediately to its left or right. What is the expected number of unpecked chicks?" This question is from the 2017 Raytheon National MathCounts competition that took place today. It took me like 5 minutes to solve it. Congrats to Luke Robitaille who buzzed in under one second and gave the correct answer! Watch the video for a detailed solution and a pedantic discussion of the wording of the question.

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[MindYourDecisions]

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1. If there's a "hard" problem or any math topic you want covered, please let me know. Preferably send me an email--my address is listed on my blog and in the channel's "about" tab. I do extensive research so it takes me months to make some videos. The Google 25 horses puzzle, for example, was from an email I got in October 2016.

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[vibhav]

I solved it in 1 second too, I just wasn't right.

[PavlosPapageorgiou]

Zero. There will be mayhem in the barn within a few seconds.

[stingersfury]

to simplify the long maths: Each chicken has 2 chickens total, 1 to its left and 1 to its right right which each have a 50% chance of pecking it. You take a 50% chance twice, you have 25% chance of not getting pecked per chicken which leaves an expected 25 chickens left unpecked total. It’s basically just 2 coin flips and asking what’s the expected chance of it not hitting tails at least once. This was probably the way the 13 year old did it in his head quickly.

[rayrowley]

For any given chicken there are 4 options: Pecked by left, pecked by right, pecked by both, pecked by neither. As you say we assume each scenario is equally likely so the answer is 25% or 25 chickens.

[anubhabghosal]

I solved it in 1 second too. That's because I have seen this problem previously and not because I am some sort of genius.

[oriontaylor]

This is a great example of how powerful linearity of expectation is. It allows you to analyze the random variables for whether or not the ith chick was pecked individually, despite the fact that the random variables are all correlated.

[IsaSaien]

when a math problem says "a kid did it in one second" that translates into "this is an intuiton problem disguised as a math problem"

[neelnarlawar]

Mother hen: how did you get hurt?
Chick: It's complicated

[Bai_Su_Zhen]

I missed they were in a circle so I got confused about dealing with the ones on the edge lol

Anyways to explain easily, there are 4 possible outcomes for each chick. Either it's pecked from the right, or from the left, or not at all, or pecked 2 times. Since it's random, there is 0.25 chance to all of these. So there is 0.25 chance for each chick not to be pecked. 100*0.25=25

[cpm]

Every chick has to peck another chick. Sure as hell that chick is going to peck back in revenge. So they'd all get pecked at least once. This is known as the Newton's 3rd Law solution.

[irok]

But it doesn't matter because of quantum physics. Each chicken pecks both ways and is pecked from both sides before the event is observed

[DaemonJax]

If you've seen this problem before, you know you just need to divide by 4. It all comes down to access to preparation and how much that preparation costs (in dollars and time).

[blue_tetris]

Why are all the titles like "90% of college students got this wrong" or "The Chinese can answer this! Can you?" or "4 out of 5 stay-at-home Peruvian college dads can't get this wrong. Do you?"

These puzzles are fun and thought-provoking, but there's a Facebooky clickbaity bullshizz title that rolls my bespectacled fuggin' eyes unwillfully out my dang skull every time. Just solve me the damned puzzle, dogg.

[noelleggett]

I think any chick that can manage to flip a coin or roll a die deserves the scholarship!

[johnhofmann]

Got it pretty quick. Each chicken has two independent 50% chances of being pecked, which means a 75% chance of being pecked, which means 25% chance of being unpecked.

[Prashik_ft]

Who said they could peck only once?
So zero is an answer too

[morningtarr]

So I did this practically,
Now i came to conclusion that

The Chicks can't seat peacefully.

[jffrysith]

I thought about it from a single chicken perspective, assume you just look at one chicken. (As chances are unstated, I assumed the odds to be 50%.)
From the perspective of a single chicken, for them to be pecked, they must be pecked by either the left chicken, the right chicken or both chickens. As the chances of being pecked by one of either chicken is 50%, the chances of not being pecked by 1 of either chicken is also 50%. Therefore, since the chance of being pecked by the other is independent to whether it was pecked by the first chicken, we can simply multiply the chances to get P(unpecked) = 0.5 * 0.5 = 0.25 or 25%. Since there are 100 chickens, this will leave 25 unpecked chickens on average.

[Pumbear]

Situation:
Roll a six-sided die, what's the chance you will roll a 6?

Normal answer:
1/6

mind your decision's answer:
Well the die could be weighted so who fucking knows? And where are you rolling it? In outer space? You can't roll anything when the die doesn't land. And what is even on the sides of the die? If it's all sixes then surely the chance to roll a six is 6/6. I mean if they don't literally describe every atom of the die there is simply no way to say for sure the chance will be 1/6!