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Suppose you have a deck of 52 playing cards, fully shuffled, one of which is the King of Hearts. You perform this procedure: 1. Put the top three cards of the deck into a second pile. If there aren't three cards, put the rest of the deck into the pile. 2. If the King of Hearts in the second pile, shuffle the pile and the deck together.

What is the expected position of the King of Hearts in the deck after repeating this process an arbitrarily large number of times? What's the general approach to this problem with N cards in the deck and M cards revealed every time through?

(This actually came up as the result of a Magic: the Gathering scenario, but I assume that playing cards are more familiar.)

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# Expected position of a card in a deck after repeating a procedure

Suppose you have a deck of 52 playing cards, one of which is the King of Hearts. You perform this procedure: 1. Put the top three cards of the deck into a second pile. If there aren't three cards, put the rest of the deck into the pile. 2. If the King of Hearts in the second pile, shuffle the pile and the deck together.

What is the expected position of the King of Hearts in the deck after repeating this process an arbitrarily large number of times? What's the general approach to this problem with N cards in the deck and M cards revealed every time through?

(This actually came up as the result of a Magic: the Gathering scenario, but I assume that playing cards are more familiar.)