Suppose I have a deck of 52 cards, which I have numbered from 1 to 52. I then apply the following shuffling algorithm on the deck -
- Start from the first card. Then choose a random number from 1 to 52, and swap the first card with the card at the position of that number. So for example if the random number is 50, we swap the card with 1 and the card at position 50.
- Then move onto the card with 2 on it. Again select a random number between 1 and 52 and swap these two cards. Note that the card with 2 may get swapped with the cards before it as well, not only the ones after it.
- Continue this process until we reach the end of the deck.
Now I know that this will not produce a uniform distribution of the cards,e since we are also swapping the cards with ones behind them. What I am interested in is the most probable and least probable permutation of the cards obtained after such a shuffling. Can anyone tell me how to proceed on this problem? Thanks in advance.