An intriguing probability problem. At a birthday party n children are given hats. The hats are later shuffled and randomly reassigned. The probability that exactly k get the same hat twice is . As n increases, this probability approaches (1/k!)e−1. The probability that none gets the same hat twice approaches e−1≈0.368. Note that the result for n=5 is 0.367, so that the probability of no matches is essentially the same whether there are 5 or 500 children.
Subjects: Probability and Statistics.