There's no reason to require prime numbers, the game works the same with picking the lowest unique positive integer. We actually played this game at our department's research symposium two years ago with about a hundred people. Nobody picked 5 and the winning entry was 11.
Actually, there is a reason. It forces people (most people, at any rate), to actually look at a list of numbers, rather than simply thinking of one.
Like, imagine you asked a group of people to pick a number between 1-100. Some numbers are going to occur out of proportion to a random distribution. If it's a geeky crowd, you'll probably get at least 10% picking 42.
Whereas if you give them a list of the first 100 primes, and ask them to pick one, that sort of cultural bias should be mostly gone.
I realize that's not exactly the scenario involved here, but the concept still applies imo.
"Some numbers are going to occur out of proportion to a random distribution. If it's a geeky crowd, you'll probably get at least 10% picking 42."
This is so true that, for many years, magicians would use it as a "mind reading" parlor trick. The number 7 comes up with outsized frequency when people are asked to choose a number between 1 and 10 (the further out you expand the set, the lower the probability of their picking 7 -- but not by as much as you might expect). So much so, that you could fairly reliably ask someone to pick a number, guess that they'd picked 7, and be correct.
This is because most people don't actively think about the answer. They just select the number that comes most readily to mind. In Western culture, 7 is ingrained fairly heavily as a lucky/special number, and we encounter it with enough frequency that it'll be a likely choice for the brain's equivalent of the auto-fill feature.
You're much more likely to get a truly random sampling if your question requires thought or calculation, or if the bounds are unusual enough to warrant active thought. For instance, "Pick a number between 2 and 99" isn't drastically different from "Pick a number between 1 and 100." But it's different enough that it cues the brain to stop for a millisecond and actually think about the question. As a result, it's slightly more likely to generate a unique/uncommon number. (An even stranger set, such as "Pick a number between 6 and 73," will force the person to think even longer).
