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A special case.

Despite the surprisingly simple shape of

, it still appears intractable to find a general closed form for the eigenvalues of M as a function of m. We therefore work out an approximate closed form for the case of the standard deck with m=10 face values. Since we are approximating the eigenvalues in this section, the expression we derive is only valid up to a pre-assigned number of significant digits, which we pick to be 4.

Using the characteristic polynomial from the previous section, and invoking the Maple symbolic manipulation program, we find that the eigenvalues of M are, up to 4 significant digits, as follows: , and . Again using Maple, we can diagonalize M and exponentiate it to get a closed form for . Reading off the first entry of this vector gives the probability that the dealer's k'th secret card will coincide with one of the player's secret cards:

It is evident that only the first couple of eigenvalues, namely 1 and , control the behavior of the probability for large k. Also note that we now see explicitely the exponential growth of the probability . The following table compares an actual simulation to the theory developed. The first column denotes the card number in the deck, which is the dealer's k'th secret card times the expected face value of a card ( in our case). The second column is the dealer's k'th secret card, the third column gives the simulation probabilities on 10,000 runs with random initial seeds, and the fourth column gives the predicted probabilities derived above from the Markov process.

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