An energetic boy got a piggy bank for his birthday. He decided that from then on he will number every bill he gets from his grandparents (1, 2, …) and put it all in his bank. During the first half year he got 2 bills, but at the end of this period he pulled out 1 bill (chosen at random). In the next 1/4 year he got 2 more bills, but at the end of this period he pulled 1 bill chosen at random from the 3 bills in his bank. In the next 1/8 year he repeated the same routine etc. (each period is half the length of the previous period). What is the probability that any of the bills he got during this year will remain in his bank after a full year of the above activity? Paradoxically the probability is 0, even though it is clear that he only spent half of his money. Can we offer the boy good financial advice without making him cut his expenses?
— Talma Leviatan, “On the Use of Paradoxes in the Teaching of Probability,” Proceedings of ICOTS 6, 2002