Oddly enough when I saw this year's final question, I almost immediately said out loud 25% of the total number or 25. Somewhere very recently I'd seen this problem (I can't remember exactly where), I didn't recall the reasoning offhand but the answer came to mind instantly. Math Counts at the national level works a bit like that. Seeing lots of problems and being able to quickly either recall the entire answer or the efficient means to solve it is critical to win where kids are answering questions like above in a few seconds. In fact, I couldn't even "borrow" some of the questions since they were answered before being fully read out or printed on the screen.
That's not super interesting in the long run, but I think its balanced out by what happens when the larger set of kids are preparing for the contest and even when bystanders read an article in the nytimes and spend some time thinking about a problem.
This one is fairly fun to model. Many people eventually came to the reasoning that for an individual chick there is a 25% chance it won't be pecked. But this is not independent of what happens to all the other chicks. Because this chick wasn't pecked 2 other neighbors were. I like thinking about this as a chain of Ls and Rs where your counting the number of transitions between letters.
Behind all of this is the somewhat counter intuitive: Linearity of Expectation. Even though the individual outcomes are dependent the expected value can be had by simply adding them up anyway.
I'm planning to do a session around this today divorced from time pressures. Building up to the Math Counts question through a series of exercises and observations about expected value should make an excellent white board #vnps activity.
Followup interesting tweet stream on the probability distribution:
@stevenstrogatz @SimonDeDeo Late-night coding error. Sorry! Turns out, not binomial. Correct simulation shows much smaller variance, as you predicted. pic.twitter.com/yXN0YaIbpt— Aaron Clauset (@aaronclauset) May 16, 2017