megan poole senior talk

The Mathematics of MagicVariation Distances and the Cutoff Phenomenon in Card Shuffling

Megan Poole

Senior TalkFebruary 24, 2015

Megan Poole Card Shuffling

History


Types of Shuffles

Perfect Shuffle

Perfect interleaving of a deck.

Top-in-at-Random-Shuffle

Top card inserted into the deck at random.

Riffle Shuffle

Random interleaving of two halves of a deck.


To begin...

We must know:

Deck order

Deck size

Type of Shuffle

Measure of Randomness


Top-in-at-Random Shuffle


Measure of Randomness: Variation Distances

d(k) ≡ ‖Qk∗ − U‖ → 0 as k →∞


Variation Distances and the Cutoff Phenomenon

Prove:

d(n log n + cn) ≤ e−c ; all c ≥ 0, n ≥ 2

d(n log n − cnn)→ 1 as n→∞; all cn →∞


Coupon Collector’s Problem

Problem

Given n coupons, how many coupons do you expect you need todraw with replacement before having drawn each coupon at leastonce?


Coupon Collector’s Problem

PROOF


Proving Theorem 1

Prove:

d(n log n + cn) ≤ e−c ; all c ≥ 0, n ≥ 2

d(n log n − cnn)→ 1 as n→∞; all cn →∞


Basic Math of Card Shuffling

GSR Model of Shuffling

Cut: Binomial Distribution

Shuffle: Probability determined by number of cards in eachhalf of the cut

Cutoff Phenomenon: 3/2 log2 n

Rising Sequences


Rising Sequences

Rising Sequences


Applications

Magic Tricks!

Gambling and Card Counting


Questions?


References

M. Aigner & G. Ziegler: Proofs from the book, (4th ed.). Berlin:Springer. (2010), 185-194.

D. Aldous & P. Diaconis: Shuffling cards and stopping times,Amer. Math. Monthly 93 (1986), 333-348.

D. Bayer & P. Diaconis: Trailing the dovetail shuffle to its lair,Annals Applied Probability 2 (1992), 294-313.


megan poole senior talk

Documents

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dn log n cn ec

dovetail shue

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