I have a deck of 52 cards on the table in front of me, in random order, each one randomly facing either up or down. I place one joker at each end, each of these again randomly facing either up or down, to get a deck of 54 cards.
I then perform a sequence of moves each of the following form.
Choose a random contiguous block of cards (a certain number $\geq1$ of cards, all together in the deck) such that both the top and bottom card are facing up, and flip it over as a single unit.
What is the probability that eventually I can no longer perform such a move, i.e. all the cards are facing down?