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Ken's Puzzle of the Week

Cards in Order

A deck of N cards is numbered 1 to N and shuffled. If dealt from top to
bottom, what is the probability of dealing at least one pair of successive cards
in their proper order (i.e a 4 followed by a 5, at any position in the deck.)
For N=2, 3, 4, the probabilities are 1/2, 1/2, 13/24. Solve for N up to 10
(or higher). Does the probability ever decrease or is there a limit as N
increases?

Extension: What is the expected number of such successive pairs in an N card
deck? (Pairs may overlap: 52341 has two successive pairs - 23 and 34.)

Source: Mark Moyer from UVM, citing practical
experience with the random order of submitted homework assignments.

Solution

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