Calculating the Expected Time for a Deck of 52 Cards to Return to Its Original Order

Imagine you have a deck of 52 cards, numbered from 1 to 52. Every second, you perform a shuffle by drawing a card at random and placing it on top of the deck. How long would you have to wait, in expected time, until this deck returns to its original order? This problem can be approached using concepts from probability and expected values, particularly through the lens of a Markov chain. Let's delve into the details.

Key Concepts

Random Shuffle

Each shuffle involves selecting one card from the deck and placing it on top, which changes the order of the cards. This random process is the core of our problem.

States of the Deck

The deck can be in any one of 52! (52 factorial) different states or arrangements. This vast number of states makes the calculation both theoretically interesting and practically challenging.

Markov Chain

The process can be modeled as a Markov chain where each state corresponds to a unique arrangement of the cards. A Markov chain is a stochastic process where the future state depends only on the current state and not on the past states.

Step-by-Step Calculation

To calculate the expected time until a deck of 52 cards returns to its original order, we need to use the concept of expected time in a Markov chain. The expected time to return to the initial state in a Markov chain is given by the formula:

Expected Time to Return to a State

E frac{T}{pi_i}

E is the expected time until the Markov chain returns to its initial state. T is the total number of states, which for a deck of 52 cards is 52!. pi_i is the stationary distribution probability of being in state i, which in this case is the initial state.

Markov Chain Characteristics

In a uniform random process, the stationary distribution is uniform across all states. This means:

(pi_i frac{1}{T} frac{1}{52!})

Expected Time Calculation

Substituting pi_i into the expected time formula, we get:

E frac{52!}{frac{1}{52!}} 52!^2

So, the expected amount of time until the deck returns to its original order is 52!^2 seconds.

Note on Practicality

While the formula provides a theoretical answer, calculating 52! directly is impractical due to its enormous size, approximately 8.0658 times 10^{67}. The sheer scale of the numbers involved makes this calculation theoretically interesting but practically infeasible.

Therefore, while the formula gives a theoretical answer, it is primarily of academic interest. In practice, the time it would take for a deck of cards to return to its original order through this random shuffle is astronomically large.