TITLE:
Progressive Randomization of a Deck of Playing Cards: Experimental Tests and Statistical Analysis of the Riffle Shuffle
AUTHORS:
M. P. Silverman
KEYWORDS:
Randomization of Cards, Number of Riffle Shuffles, Rising Sequences, GSR Model, Entropy and Information
JOURNAL NAME:
Open Journal of Statistics,
Vol.9 No.2,
April
28,
2019
ABSTRACT: The
question of how many shuffles are required to randomize an initially ordered
deck of cards is a problem that has fascinated mathematicians, scientists, and
the general public. The two principal theoretical approaches to the problem,
which differed in how each defined randomness, has led to statistically different
threshold numbers of shuffles. This paper reports a comprehensive experimental
analysis of the card randomization problem for the purposes of determining 1)
which of the two theoretical approaches made the more accurate prediction, 2)
whether different statistical tests yield different threshold numbers of
randomizing shuffles, and 3) whether manual or mechanical shuffling randomizes
a deck more effectively for a given number of shuffles. Permutations of 52-card
decks, each subjected to sets of 19 successive riffle shuffles executed
manually and by an auto-shuffling device were recorded sequentially and
analyzed in respect to 1) the theory of runs, 2) rank ordering, 3) serial
correlation, 4) theory of rising sequences, and 5) entropy and information theory.
Among the outcomes, it was found that: 1) different statistical tests were
sensitive to different patterns indicative of residual order; 2) as a
consequence, the threshold number of randomizing shuffles could vary widely
among tests; 3) in general, manual shuffling randomized a deck better than mechanical shuffling for a given
number of shuffles; and 4) the mean number of rising sequences as a function of
number of manual shuffles matched very closely the theoretical predictions
based on the Gilbert-Shannon-Reed (GSR) model of riffle shuffles,
whereas mechanical shuffling resulted in significantly fewer rising sequences
than predicted.