probability and random processes - siit.tu.ac.th - 4 - combinatorics u3.pdf · asst. prof. dr....

Asst. Prof. Dr. Prapun [email protected]

4 Combinatorics

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Probability and Random ProcessesECS 315

Office Hours: BKD, 6th floor of Sirindhralai building

Tuesday 9:00-10:00Wednesday 14:20-15:20Thursday 9:00-10:00

Supplementary References

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Mathematics of Choice How to count without counting By Ivan Niven permutations, combinations, binomial

coefficients, the inclusion-exclusion principle, combinatorial probability, partitions of numbers, generating polynomials, the pigeonhole principle, and much more

A Course in Combinatorics By J. H. van Lint and R. M. Wilson

Cartesian product

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The Cartesian product A × B is the set of all ordered pairs where and .

Named after René Descartes

His best known philosophical statement is “Cogito ergo sum”(French: Je pense, donc je suis; I think, therefore I am)

[http://mathemagicalworld.weebly.com/rene-descartes.html]

Heads, Bodies and Legs flip-book

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Heads, Bodies and Legs flip-book (2)

5

Interactive flipbook: Miximal

6 [http://www.fastcodesign.com/3028287/miximal-is-the-new-worlds-cutest-ipad-app]

One Hundred Thousand Billion Poems

7 [https://www.youtube.com/watch?v=2NhFoSFNQMQ]

One Hundred Thousand Billion Poems

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Cent mille milliards de poèmes

Pokemon Go: Designing how your avatar looks

9

How many chess games are possible?

10 [https://www.youtube.com/watch?v=Km024eldY1A]

An Estimate: The Shannon Number


The Shannon Number: Just an Estimate


Example: Sock It Two Me

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Example: Sock It Two Me

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Jack is so busy that he's always throwing his socks into his top drawer without pairing them. One morning Jack oversleeps. In his haste to get ready for school, (and still a bit sleepy), he reaches into his drawer and pulls out 2 socks.

Jack knows that 4 blue socks, 3 green socks, and 2 tan socks are in his drawer.

1. What are Jack's chances that he pulls out 2 blue socks to match his blue slacks?

2. What are the chances that he pulls out a pair of matching socks?

[Greenes, 1977]

“Origin” of Probability Theory

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Probability theory was originally inspired by gamblingproblems.

In 1654, Chevalier de Mere invented a gambling system which bet even money on case B.

When he began losing money, he asked his mathematician friend Blaise Pascal to analyze his gambling system.

Pascal discovered that the Chevalier's system would lose about 51 percent of the time.

Pascal became so interested in probability and together with another famous mathematician, Pierre de Fermat, they laid the foundation of probability theory.

best known for Fermat's Last Theorem

[http://www.youtube.com/watch?v=MrVD4q1m1Vo]

Example: The Seven Card Hustle

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Take five red cards and two black cards from a pack. Ask your friend to shuffle them and then, without looking at the

faces, lay them out in a row.

Bet that them can’t turn over three red cards. The probability that they CAN do it is

[Lovell, 2006]

3

3

5 57

4 3

7 6 5

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53 5!7 3!3

3!2!

4! 1 25 4 37! 7 6 5 7

Permutation

19[https://www.youtube.com/watch?v=2EN6jYMiGF4]

Permutation

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There are 11! = 39,916,800 ways to permute 11 persons.

[https://www.youtube.com/watch?v=2EN6jYMiGF4]

Finger-Smudge on Touch-Screen Devices

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Fingers’ oily smear on the screen

Different apps gives different finger-smudges.

Latent smudges may be usable to infer recently and frequently touched areas of the screen--a form of information leakage.

[http://www.ijsmblog.com/2011/02/ipad-finger-smudge-art.html]

For sale… Andre Woolery Art

22[http://www.andrewooleryart.com/all-artwork/]

Andre Woolery Art

23

Fruit Ninja Facebook

Angry Bird Mail

Lockscreen PIN / Passcode

24[http://lifehacker.com/5813533/why-you-should-repeat-one-digit-in-your-phones-4+digit-lockscreen-pin]

Smudge Attack

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Touchscreen smudge may give away your password/passcode

Four distinct fingerprints reveals the four numbers used for passcode lock.

[http://www.engadget.com/2010/08/16/shocker-touchscreen-smudge-may-give-away-your-android-password/2]

Suggestion: Repeat One Digit

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Unknown numbers: The number of 4-digit different passcodes = 104

Exactly four different but known numbers: The number of 4-digit different passcodes = 4! = 24

Exactly three different but known numbers: The number of 4-digit different passcodes = 23 4 36

Choose the number that will be repeated

Choose the locations of the two non-repeated numbers.

50% improvement

News: Most Common Lockscreen PINs

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Passcodes of users of Big Brother Camera Security iPhone app

15% of all passcode sets were represented by only 10 different passcodes

[http://amitay.us/blog/files/most_common_iphone_passcodes.php (2011)]

out of 204,508 recorded passcodes

Even easier in Splinter Cell

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Decipher the keypad's code by the heat left on the buttons.

Here's the keypad viewed with your thermalgoggles. (Numbers added for emphasis.) Again, the stronger the signature, the more recent the keypress.

The code is 1456.

Actual Research

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University of California San Diego The researchers have shown that codes can be easily discerned

from quite a distance (at least seven metres away) and image-analysis software can automatically find the correct code in more than half of cases even one minute after the code has been entered.

This figure rose to more than eighty percent if the thermal camera was used immediately after the code was entered.

K. Mowery, S. Meiklejohn, and S. Savage. 2011. “Heat of the Moment: Characterizing the Efficacy of Thermal-Camera Based Attacks”. Proceed-ings of WOOT 2011.http://cseweb.ucsd.edu/~kmowery/papers/thermal.pdfhttp://wordpress.mrreid.org/2011/08/27/hacking-pin-pads-using-thermal-vision/

Alternative Solution

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Scramble PIN Layout in CyanogenMod (a modified version of Android )

Scramble the keypad when you unlock your phone so that people can't peek at your keystrokes and learn your PIN.

The Birthday Problem (Paradox)

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How many people do you need to assemble before the probability is greater than 1/2 that some two of them have the same birthday (month and day)? Birthdays consist of a month and a day with no year attached. Ignore February 29 which only comes in leap years Assume that every day is as likely as any other to be someone’s

birthday

In a group of r people, what is the probability that two or more people have the same birthday?

The Birthday Problem (con’t)

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With 88 people, the probability is greater than 1/2 of having three people with the same birthday.

187 people gives a probability greater than1/2 of four people having the same birthday

[Rosenhouse, 2009, p 7][E. H. McKinney, “Generalized Birthday Problem”: American MathematicalMonthly, Vol. 73, No.4, 1966, pp. 385-87.]

Birthday Coincidence: 2nd Version

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How many people do you need to assemble before the probability is greater than 1/2 that at least one of them have the same birthday (month and day) as you?

In a group of r people, what is the probability that at least one of them have the same birthday (month and day) as you?

Permuting four distinct letters: A,B,C,D

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DCBA

DCABDBCADBACDABC

DACB

CDBA

CDABCBDACBADCABD

CADB

BCDA

BCADBDCABDACBADC

BACD

ACBD

ACDBABCDABDCADBC

ADCB

Permuting four letters: A,A,B,C

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ACBA

ACABABCAABACAABC

AACB

CABA

CAABCBAACBAACABA

CAAB

BCAA

BCAABACABAACBAAC

BACA

ACBA

ACABABCAABACAABC

AACB

D

ACBA

ACABABCAABACAABC

AACB

CABA

CAABCBAACBAACABA

CAAB

BCAA

BCAABACABAACBAAC

BACA

ACBA

ACABABCAABACAABC

AACB

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Permuting four letters: A,A,B,CD

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DCBA

DCABDBCADBACDABC

DACB

CDBA

CDABCBDACBADCABD

CADB

BCDA

BCADBDCABDACBADC

BACD

ACBD

ACDBABCDABDCADBC

ADCB


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DCBA

DCABDBCADBACDABC

DACB

CDBA

CDABCBDACBADCABD

CADB

BCDA

BCADBDCABDACBADC

BACD

ACBD

ACDBABCDABDCADBC

ADCB


Group 1

Group 2

Distinct Passcodes (revisit)

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Unknown numbers: The number of 4-digit different passcodes = 104

Exactly four different numbers: The number of 4-digit different passcodes = 4! = 24

Exactly three different numbers: The number of 4-digit different passcodes =

Exactly two different numbers: The number of 4-digit different passcodes =

Exactly one number: The number of 4-digit different passcodes = 1

Check:

104 ⋅ 24 10

3 ⋅ 36 102 ⋅ 14 10

1 ⋅ 1 10,000

23 4 36

43

42

41 14

Ex: Poker Probability

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[ http://en.wikipedia.org/wiki/Poker_probability ]Need more practice?


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[http://www.wolframalpha.com/input/?i=probability+of+royal+flush]


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[http://www.wolframalpha.com/input/?i=probability+of+full+house]


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[http://www.wolframalpha.com/input/?i=probability+3+queens+2+jacks&lk=3]

Binomial Theorem

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1 1 2 2( ) ( )x y x y

1 2 1 2 1 2 1 2x x x y y x y y

1 1 2 2 3 3( ) ( ) ( )x y x y x y

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3x x x x x y x y x x y y y x x y x y y y x y y y

( ) ( )x y x y

( ) ( ) ( )x y x y x y

2 323 33xyy yxy yyxx yx

xyy xyx y yy

yxxxx

xxx y

2 22yyxx xxy yx xy y

1 2 3

1 2 3

x x xy yy

xy

How many chess games are possible?: A Revisit


Sequence A048987

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Number of possible chess games at the end of the n-th ply (move). Often called perft(n) The name “perft” comes from a command in the chess

playing program Crafty.

https://oeis.org/A048987

0 1

1 20

2 400

3 8902

4 1972815 4865609

6 119060324

7 3195901860

8 84998978956

9 2439530234167

10 69352859712417

11 2097651003696806

12 62854969236701747

13 1981066775000396239

[http://wismuth.com/chess/statistics-games.html]

(16 pawn moves and 4 knight moves)

It all began at Cornell…

55 [https://www.youtube.com/watch?v=vKPRRUWbWAQ]

IEEE Richard W. Hamming Medal

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1988 - Richard W. Hamming1989 - Irving S. Reed1990 - Dennis M. Ritchie and Kenneth L. Thompson1991 - Elwyn R. Berlekamp1992 - Lotfi A. Zadeh1993 - Jorma J. Rissanen1994 - Gottfried Ungerboeck1995 - Jacob Ziv1996 - Mark S. Pinsker1997 -Thomas M. Cover1998 - David D. Clark1999 - David A. Huffman2000 - Solomon W. Golomb2001 - A. G. Fraser

2002 - Peter Elias2003 - Claude Berrou and Alain Glavieux2004 - Jack K. Wolf2005 - Neil J.A. Sloane2006 -Vladimir I. Levenshtein2007 - Abraham Lempel2008 - Sergio Verdú2009 - Peter Franaszek2010 -Whitfield Diffie, Martin Hellman and Ralph Merkle2011 -Toby Berger2012 - Michael Luby, Amin Shokrollahi2013 - Arthur Robert Calderbank2014 -Thomas Richardson and Rüdiger L. Urbanke

For contributions to coding theory and its applications to communications, computer science, mathematics and statistics.

[http://www.ieee.org/documents/hamming_rl.pdf]

Neil J.A. Sloane

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British-U.S. mathematician. Received his Ph.D. in 1967 at Cornell University. Joined AT&T Bell Labs in 1968 and retired from AT&T Labs in

2012. Major contributions are in the fields of combinatorics, error-

correcting codes, and sphere packing. Best known for being the creator and maintainer of the On-Line

Encyclopedia of Integer Sequences.

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How good was the Shannon’s estimation?

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0 2 4 6 8 10 12 14100

105

1010

1015

1020

#moves

Actual number of possible gamesShannon's estimates

It is quite easy to get a better approximation

59

0 2 4 6 8 10 12 14100

105

1010

1015

1020

#moves

Actual number of possible gamesShannon's estimatesPrapun's estimates

probability and random processes - siit.tu.ac.th - 4 - combinatorics u3.pdf · asst. prof. dr....

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