Adv Probability Question

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You have a set of 52 cards, where each card comes in one of 4 distinct colors (blue, green, red, yellow) and has written on it one of 13 distinct values (the integers from 1 through 13, inclusive), with each value0color pair appearing exactly once. You lay all 52 cards in a single row in some random order. The “distinct double count” is defined as the maximum number of non-overlapping consecutive pairs of cards, where each pair has the same number written on both cards. (For example, the sequence “3 3 3” has a distinct double count of 1, and the sequence “3 3 3 3” has a distinct double count of 2). The expected value of the distinct double count can be written in the form A/B, where A and B are relatively prime integers. Compute A + B.

 

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