We shuffle a deck by the following method:

*Take the top card off the top of the deck and place back into the deck in a position chosen uniformly at random.*

It is allowable that the top card will be put directly on top again.

Let denote the position of the Queen of Hearts at time . We assume that the Queen of Hearts starts on top, i.e. , and wish to compute the expected time for it to return to the top.

The probability transition matrix when there are a total of FIVE cards is:

*What is the expected time until the shuffler sees the Queen of Hearts on top of the deck when there are 52 cards?*

You may use the book, but you may not communicate with other students on this problem. Work smart! The solution does not require solving a 52-dimensional system of equations. (Hint: It may help to solve the 5-card case first and then attack the larger problem.)