If you shuffle n cards as follows: Go through the deck one card at a time and at each card, flip a fair coin. If the coin comes up head, then leave the card where it is, and if it comes up tails move that card to the end of the deck. For example, if n =4, and the initial ordering is 1,2,3,4, and outcome of the successive flips is h,t,t,h, then the ordering at the end of the round is 1,4,2,3. Assuming all possible outcomes of the coin flips are equally likely, what is the probability that the ordering after one round is the same as the initial ordering?
If the ordering after n flips is the same as the initial ordering, then you would have to roll all heads since rolling heads doesn't change the ordering. So the answer would be (1/2)^n?