The probability of winning is exactly 1/13.
Consider how the game could end. Could it end by you picking up a card, and going to the corresponding pile, and finding no card there? It seems unlikely for this to happen. For instance, how could you come to the 7 pile, and have nothing to pick up there? This could happen only if all four cards that had been on the 7 pile at the start were now gone. How could they be gone? Only because you had encountered a 7 four times already, and each time taken a card. But wait, if the four cards in the 7 pile are already gone because you encountered four 7's already, then how did you get back to the 7 pile a fifth time? You can't! You will never come to the 7 pile and not find a card there for you.
Doesn't this argument work for any pile, meaning you can't lose? Well, it almost does. The same argument is true for all piles but the King pile. The King pile is the only pile from which we have taken one card in an unusual way. Instead of picking up a king somewhere and coming to the King pile, we just started there. In other words, then, it is possible to come to the King pile, and for there to be no card there. This would happen when the 4th King is encountered. And as a drastic example, simply suppose the King pile had all four kings in it. Then the game is over after four draws, and you lose!
To see better what's going on, forget about the fact that the cards are in piles. We'll make up a new but completely equivalent game called "Score Four". Simply take a randomly shuffled deck, turn over one card at a time, and build piles for each rank. Give the kings a head start by stealing a king from another deck and using it to start the king pile. Stop the game as soon as you have five cards in any pile.
This is the same game as Clock Solitaire, but now it's obvious that "Score Four" can only terminate when the fourth king in the deck is turned over, and it terminates with a win if and only if that fourth king is the last card in the deck.
Hence, the probability of winning Clock Solitaire is exactly 1/13.
Reference:
Back to The Clock Solitaire Puzzle.