How does the trick work? Well, the dealer picks her own secret card to begin the process in her mind, say . She now carries out the same procedure, keeping track of her secret cards , etc. She keeps dealing for a while and then points out one of her own secret cards. Probabilistically, her sequence and the player's sequence will dove tail into the same sequence. The main point is that as soon as one of her secret cards coincides with one of the player's secret cards, their two sequences of secret cards are identical from that point onward.
A natural question is: ``what is the probability that the dealer will correctly guess one of the player's secret cards after n cards have been dealt?'' We reduce the problem to a discrete, absorbing Markov Process. In general, we find the m by m matrix of transition probabilites for a ``deck'' of cards with m face values. In the last section we go back to the standard deck with m=10 face values, where we throw out the jacks, queens, and kings for simplicity.
The authors would like to thank Jeff Lagarias for pointing out an
interesting preprint  which uses somewhat different methods.
Martin Gardner has also discussed this card trick in his column
of the Scientific American .