I had a math teacher who trolled the entire class.
We (the class) wanted to go outside instead of having a regular lecture. He made a deal that if we win his game we go outside, otherwise we had to be extra focused for the rest of the period.
Everyone in the class had to pick a random number between 1 and 100 and write it down. He told us that he would get one guess for each pupil (around 30) and if he managed to guess all numbers he would win. If any number was left we go outside.
Knowing how bad humans (and especially 12 year olds) are at picking random numbers, and knowing which numbers were mostly picked, he guessed all numbers with still a handful of guesses left.
If he had done it enough times with different kids in the past, he would have the perfect data to know the likelihood of certain numbers picked and him winning.
That from a teacher's perspective seems like a fascinating idea to try out.
I think it's much better because the teacher doesn't have to match guesses to students. For example, for each student the odds are 30/100, roughly one in three. And any duplicates can be matched by a single guess.
That formula is missing something because for over 100 students the teacher can't lose. (they get over 100 guesses and there are only 100 numbers possible)
More precisely, if there are N students, the probability is (min(N,100)/100)^N. This is 1 for N ≥ 100. And the probability at N=30 is indeed a tiny 2e-16, which shows that the children's "random" picks were far from uniformly random.
(Incidentally, even with N=99 the probability is 0.37 ≈ 1/e, and the probability is lowest at N=37 ≈ 100/e. This is not a coincidence.)
The teacher picks 30 numbers out of 100. Then each student (independently) picks one number. If random, that is 0.3 ^ 30. Obviously, the students are not picking random.
We (the class) wanted to go outside instead of having a regular lecture. He made a deal that if we win his game we go outside, otherwise we had to be extra focused for the rest of the period.
Everyone in the class had to pick a random number between 1 and 100 and write it down. He told us that he would get one guess for each pupil (around 30) and if he managed to guess all numbers he would win. If any number was left we go outside.
Knowing how bad humans (and especially 12 year olds) are at picking random numbers, and knowing which numbers were mostly picked, he guessed all numbers with still a handful of guesses left.