52-pickup is the practical joke "game" where the mark finds out that the game consists of the proposer spraying all 52 cards of a deck into the air, letting them fall onto the floor, and then the mark has to pick them up.
Suppose all 52 cards are now on the floor, each card randomly and independently face up or down with probability 1/2.
What is the probability that the sum of the face-up cards (counting A, J, Q, K as 1, 11, 12, 13 respectively) is a multiple of 13?
Intuition does indeed argue that the answer is about 1/13. It can't be exactly 1/13, however, because the number of diffirent combinations, 2^52, is not divisible by 13.