How to quantify randomness?

Question

I am testing a thought experiment I came up with involving the relationship between the randomness within a deck of cards and the number of shuffles. The only problem is that I do not have a way to quantify "randomness." I initially thought about using the displacement of a number of cards within the deck to find the "average randomness" but that feels like it is far from an accurate representation. Is there any way to represent the amount of randomness in a deck, assuming that a non random deck has the cards ordered, from bottom to top can be represented by the following 2-Dimensional Array:

[[A,2,3,4,5,6,7,8,9,10,J,Q,K],[A,2,3,4,5,6,7,8,9,10,J,Q,K],[A,2,3,4,5,6,7,8,9,10,J,Q,K],[A,2,3,4,5,6,7,8,9,10,J,Q,K]]

Answer 1

A standard approach to this problem is to treat shuffling the $52$ cards as repeatedly applying permutations to them, and then modeling the problem as a Markov Chain - specifically a random walk on $S_{52}$. The "randomness" of the deck can be described by its distribution after some number of shuffles, where the most "random" distribution would be uniform on $S_{52}$.

See, for instance, http://statweb.stanford.edu/~cgates/PERSI/papers/aldous86.pdf

Answer 2

You might want to learn about Kolmogorov complexity. Roughly speaking, the "Kolmogorov complexity of a deck" is the length of the shortest computer program that can describe order of the cards in the deck. A completely ordered deck is easy to describe; a program might say list the cards of hearts in order from A to K, then repeat for spades, clubs, and diamonds. Now, if you have a "completely random" deck, the program can't escape from the fact that it will actually have to list the cards one by one.

Keep in mind that this way of "measuring randomness" is fundamentally different from the one presented in pwerth's answer; if you look closely, ours are actually answers to two different questions. In his answer, he's talking about sequences of operations you do on a deck, and how they might change the state of the deck; his deck is actually a random variable. In my answer, the deck is a concrete deck: an actual sequence of (concrete) cards, like "7 of hearts, 3 of spades, etc, etc".