Random Card Shuffling

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1 STAT 3011: Workshop on Data Analysis and Statistical Computing Random Card Shuffling Year 2011/12: Fall Semester By Phillip Yam Department of Statistics The Chinese University of Hong Kong

Course Information (1) Instructor: Phillip Yam Office: LSB G19 Office Hours: Monday 4:30 to 6:30 pm Email: scpyam@sta.cuhk.edu.

2 Course Information (1) Instructor: Phillip Yam Office: LSB G19 Office Hours: Monday 4:30 to 6:30 pm (2) Tutor: Peng-Fei Liu Office: LSB (3) Presentation: Date: Nov (Thursday) Time: from 3:30pm on at LSB C1 Duration: 15 to 20 mins each (4) Final Report Length: 10 to 15 pages. Deadline: 6 pm at Dec (Thursday)

3 Card Shuffling

4 Card Shuffling 1. How to design effective shuffles? 2. How many times of shuffles (a mixture those as suggested in (1)) does it take to randomize a deck of cards? 3. How to quantitatively test that the newly proposed shuffling ((1) and (2)) is likely to result in a randomized deck?

5 Permutations is a permutation as a way of rearranging the deck; e.g. 5 card deck: A composition of two consecutive permutations is still a permutation; Statistically, a shuffle Q is just a probability density on the space S n of all permutations of a deck of n cards; Notion of rising sequences in a permutation. 3 in ( )

6 After A Few Shuffling Convolution Q*Q on S n In general after k times of (different) shuffling, Q 1 * Q 2 * * Q k.

7 Top-In Shuffle Taking the top card off the deck; Reinserting the first card in any of the n positions between the n-1 cards in the remainder of the deck; Doing so randomly according to a uniform choice.

Perfect Shuffle: A Fake? (http://www.youtube.com/watch?

8 Perfect Shuffle: A Fake? ( Divide 52 cards into 2 equal piles; Shuffle by interlacing cards; Keep top card fixed (Out Shuffle); Problem (Magic trick): 8 shuffles => original order; For illustration: 4 cards, (12 34) (13 24) (12 34).

9 Perfect Shuffle: A Dynamical System Model Label the positions 0-51 Then 0->0 and 26 ->1 1->2 and 27 ->3 2->4 and 28 ->5 in general? 2x 0 x 25 f( x) 2 x 0 x 25 f ( x ) 2 x x 51 2 x x 51 Ignoring card 51: f(x) = 2x mod 51 Recall Congruences: 2x mod 51 = remainder upon division by 51

10 Perfect Shuffle: Its Order Minimum integer k such that 2 k x = x mod 51 for all x in {0,1,,51} True for x = 1! Minimum integer k such that 2 k - 1= 0 mod 51 Thus, 51 divides 2 k -1 k= 6, 2 k - 1 = 63 = 3(21) k= 7, 2 k - 1 = 127 k= 8, 2 k - 1 = 255 = 5(51)

11 Discrete Dynamical Systems First Order System: x n+1 = f (x n ) Orbits: {x 0, x 1, } Fixed Points Periodic Orbits Stability and Bifurcation Chaos! Most relevant! How can we implement this in card shuffling?

12 Riffle Shuffle Cut the deck into two packets (k, n-k) Choose k according to a binomial density (this is plausible in practice, approximately normal for large enough, say n > 30 ): Interleave the cards from each packet in any possible way, so that the cards of each packet maintain their own relative order, even if there are other cards in-between from another one.

13 Riffle Shuffle

14 Riffle Shuffle: Practical Implementation ( ) As before, cutting the deck according to the binomial density into two packets of size k and n-k. Drop a card from the bottom of one of the two packets onto a table, face down. Choose between the packets with probability proportional to packet size, i.e. if the two packets are of size p 1 and p 2, then the probability of the card dropping from the first is p 1 /(p 1 + p 2 ), and p 2 /(p 1 + p 2 ) from the second. For the first round, the probabilities would be k/n and (n-k) /n respectively. Next, with the numbers p 1 and p 2 being updated to reflect the actual packet sizes by subtracting one from the size of whichever packet had the card dropped last time. Example: from the 1st packet, and then from the 1st, 2nd, 2nd, 2nd, 1st, and so on. The probability of the drops occurring:

15 Riffle Shuffle: Calculations Probability of a cut with size k is Probability of an interleaving is Probability of a particular cut and interleaving is Remark: no information about the cut or the interleaving is indicated by this probability!

16 -Shuffles As a generalization of riffle shuffle; Cut the deck into packets of respective sizes p 1, p 2,, p with a probability given by the multinomial density; Interleave the cards from each packet in any way so long as the cards from each packet maintain their original relative order among themselves.

17 *Properties of -Shuffles A combination of an -shuffle and a -shuffle is equivalent to an - shuffle. k times riffle shuffle R (k) is the same as 2 k -shuffle The density of R (k) is For a permutation with r rising sequences Let A n,r be the number of permutations of n cards that have r rising sequences, called Eulerian numbers, the measure of closeness to uniform distribution is given by

18 Project Objectives Objective 1: Find a good shuffling method with minimum number of times. A mixture of previously mentioned? New style of shuffling? Etc. Not using Random Number Generator! You can use computing program to illustrate? Bring a deck of cards in to demonstrate by group-mates? Objective 2: Supporting arguments Theoretical study? Any probabilistic analysis, e.g. Combinatorics? Dynamical systems? Empirical investigations (using R?): empirical distributions? Monte- Carlo simulations? Objective 3: Justification Propose 2 different statistical testings of randomness; Comparative study.

19 Project Evaluation Same grade for everyone in the same group unless a table of division of labor force Presentation (40%) 15 to 20 mins for each group; Organization; Clearness; Time management; Critical thinking; Fluency in presentation, spoken English, Report (60%) 10 to 15 pages (excluding codes and data); MOST IMPORTANT: Creativity more than those commonly found in INTERNET; Innovation of new proposed method(s); Rigorous approach Proper use of statistical method of testing Submission: Send both PPT and Report to scpyam@sta.cuhk.edu.hk before 5pm on Dec Name Alice Bob Cathy Contribution 45% 45% 10%

20 References [1] Aldous, D. & Diaconis, P. (1987). Strong Uniform Times and Finite Random Walks. Advances in Applied Mathematics, 8, [2] Assaf, S., Diaconis, P. & Soundararajan, K. (2011). A rule of thumb for riffle shuffling. Ann. Appl. Probab. 21 (3), [3] Bayer, D. & Diaconis, P. (1992). Trailing the Dovetail Shuffle to its Liar. Ann. Appl. Probability, 2(2), [4] Kolata, G. (1990). In Shuffling Cards, Seven is Winning Number. New York Times, Jan. 9. [5] Scully, D. J. (2004). Perfect Shuffles Through Dynamical Systems. Mathematics Magazine, 77.

21 ~ END ~

COMBINATORICS AND CARD SHUFFLING

COMBINATORICS AND CARD SHUFFLING Sami Assaf University of Southern California in collaboration with Persi Diaconis K. Soundararajan Stanford University Stanford University University of Cape Town 11 May