*
A Reiteration of
“Shuffling Cards and Stopping Times”
* “Repeated shuffling is best treated as random walk on the
permutation group Sn. For later applications, we treat an
arbitrary finite group G. Give some scheme for randomly
picking elements of G, let Q(g) be the probability that g is
picked… the random variables X0, X1… are the random walk on
G with step distribution Q.”
* Random Walk: A path that consists of random steps- where
those steps follow some probability distribution.
*
Shuffling Cards and Stopping Times, pg 334
What is Random?
• Definition:
• The probability of all permutations is
equal (in other words, the likelihood of
the permutations follows the uniform
distribution.)
• Mathematical way of expressing:
• dQ(k) = lQk- UI→0 as k →∞
• Variation distance
• This is where the variation distance is
zero
Top in at random

 Take the top card and put it back into a
different place in the deck at random.
 For an example lets look at a deck of 3
cards and the probabilities of the
resulting shuffle.
π
Q(π)
123
0.333
132
0.000
213
0.333
231
0.333
312
0.000
321
0.000
Example Extended

Q1(π) Q2(π) Q3(π) Q4(π) Q5(π) Q6(π) Q7(π) Q8(π) Q9(π)
Q10(π
)
123
0.333
0.222
0.185
0.173
0.169
0.167
0.167
0.167
0.167
0.167
132
0.000
0.111
0.148
0.160
0.165
0.166
0.166
0.167
0.167
0.167
213
0.333
0.222
0.185
0.173
0.169
0.167
0.167
0.167
0.167
0.167
231
0.333
0.222
0.185
0.173
0.169
0.167
0.167
0.167
0.167
0.167
312
0.000
0.111
0.148
0.160
0.165
0.166
0.166
0.167
0.167
0.167
321
0.000
0.111
0.148
0.160
0.165
0.166
0.166
0.167
0.167
0.167
Tables courtesy of David Austin
http://www.ams.org/samplings/feature-column/fcarc-shuffle
Bridges = Segues?
When to stop?
• Stopping time
• A rule for when to stop shuffling
• Define as P(T>k)
• Which says: The probability that
the number of shuffles to
randomize the deck is greater
than the number of shuffles that
have been performed.
• We want P(T>k) to be close to
zero
Stopping Time
k
• ∥R −U∥≤P(T>k)
Stopping Time
• “The main purpose of this paper is to show
how upper bounds on d(k)… can be obtained
using the notion of strong uniform times.”
• Where d(k) is the difference between the
probability of a given permutation and the
uniform distribution.
• Approximately 11-12 riffle shuffles to
randomize
• They note that post hoc adjustments led to
another model that found that 7 is a sufficient
number to for a deck of 52 cards to be
randomized.
Last Comments
• Inverse sorting
• They consider it to be a “lovely new idea” which is important to how
they developed the model.
• It allows them to use models similar to ones of coin flips
• Inverse sorting is meant to simulate cutting the deck
• Steps
• Values of 0 and 1 are assigned to the cards in the deck according
to a binomial distribution.
• Cards of the same value are grouped together, maintaining their
order within the group
• The group of zeroes are placed on top of the group of 1’s
Questions
• How is random defined by Aldous and
Diaconis?
• How is this related to variation
distance?
• How many riffle shuffles are necessary
for a deck to be randomized- according
to Aldous and Diaconis?
• What is a stopping time?
• What is a random walk?