Question

Consider the following technique for shuffling a deck of n cards: For any initial ordering of the cards, go though the deck one card at a time and at each card, flip a coin with probability p of getting a head. If the coin comes up heads, then leave the card where it is; if the coin comes up tails, then move that card to the end of the deck. After the coin has been flipped n times, say that one round has been completed. For instance, if n = 4 and the initial ordering is 1,2,3,4, then if the successive flips result in the outcome H, T, T, H, then the ordering at the end of the round is 1,4, 2, 3. (a) What is the probability that the ordering after one round is the same as the initial ordering?

Answer 1

The probability of the ordering being the same after one round is (1-p)^(n-1).

Step-by-step explanation:

The probability of the ordering being the same after one round can be determined using the concept of permutations. Let's say there are n cards in the deck. The probability of keeping the first card in the same position is 1 (since it does not move). The probability of keeping the second card in the same position is 1-p (since it doesn't move with a probability of p). Similarly, the probability of keeping the third card in the same position is (1-p)^2, and so on. Therefore, the probability of the ordering being the same after one round is (1-p)^(n-1).