### How many shuffles are needed to randomize a deck of 52 cards?

##### Project Abstract

Mathematicians use Markov chains to model card shuffling. Bayer and Diaconis show that it would take seven riffle shuffles to randomize a deck of 52 cards. A key step in their justification is to evaluate the difference between the actual distribution of the card orderings after a given number of shuffles and the uniform distribution. We investigate this Markov chain convergence problem from a simulation-based approach. We first create a randomness measure for gauging how well-shuffled a deck is. Several hundred thousand random decks are generated to simulate the distribution of the randomness measures of well-shuffled decks. Then we examine the number of shuffles a certain shuffling scheme would take to reach the mean of this simulated distribution. In particular, we create two variations of the riffle shuffle and look at how many shuffles each shuffling scheme would take to randomize a deck of 52 cards.