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The Markov Process

We assume an infinite deck of cards, with equal distributions of each of the m face values. Suppose we know the dealer's and the player's sequences of special cards,

and

repectively. Define

to be the distance between the dealer's secret card

and the player's next closest secret card into the deck. Clearly

, and we define the states of our Markov process to be these m distances. The absorbing state is the distance 0 (For a good treatment of Markov Processes, see [1]).

Let be the transition matrix. If m is the number of states (i.e. face cards) we let and denote the index entries of M, in contrast with the standard convention. This enumeration of M's entries is more natural for us because it clarifies the fact that is the probability that if for some k the distance , then .