11. (The Clock Solitaire) An ordinary deck of 52 cards is well shuffled and dealt face down into 13 equal piles. The first 12 piles are arranged in a circle like the numbers on the face of a clock. The 13th pile is placed at the center of the circle. Play begins by turning over the bottom card in the center pile. If this card is a king, it is placed face up on the top of the center pile, and a new card is drawn from the bottom of this pile. If the card drawn is not a king, then (counting the jack as 11 and the queen as 12) it is placed face up on the pile located in the hour position corresponding to the number of the card. Whichever pile the card drawn is placed on, a new card is drawn from the bottom of that pile. This card is placed face up on the pile indicated (either the hour position or the center depending on whether the card is or is not a king) and the play is repeated. The game ends when the 4th king is placed on the center pile. If that occurs on the last remaining card, the player wins. The number of cards turned over until the 4th king appears determines the length of the game. Therefore, the player wins if the length of the game is 52.

(a) Find p(j ), the probability that the length of the game is j . That is, the 4th king will appear on the j th card.

(b) Find the probability that the player wins.

(c) Find the expected length of the game.