Two persons turn up cards alternately, each starting with a similar, but differently shuffled, pack of n cards. What is the probability that before all the cards have been dealt two identical cards are showing at the same time?

If the two players deal simultaneously the answer is unity minus the probability that none of the n pairs of cards is an identical pair. The problem is thus reduced to the calculation of the number of ways in which n letters can be put into n addressed envelopes without any letter going to the correct addressee. This is the classical example of derangements, and the solution can be found in Chap, IV of Whitworth's Choice and Chance, viz.