Probability and Random Processes ECS 315

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th 4 Combinatorics 1 Office Hours: BKD, 6th floor of

1 Probability and Random Processes ECS 315 Asst. Prof. Dr. Prapun Suksompong 4 Combinatorics 1 Office Hours: BKD, 6th floor of Sirindhralai building Tuesday 9:00-10:00 Wednesday 14:20-15:20 Thursday 9:00-10:00

principle, combinatorial probability, partitions of numbers, generating polynomials, the

2 Supplementary References 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 2

3 Cartesian product 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) 3 [

4 4 Heads, Bodies and Legs flip-book

5 5 Heads, Bodies and Legs flip-book (2)

6 Interactive flipbook: Miximal 6 [

7 One Hundred Thousand Billion Poems 7 [

8 One Hundred Thousand Billion Poems Cent mille milliards de poèmes 8

9 Pokemon Go: Designing how your avatar looks 9

10 How many chess games are possible? 10 [

11 An Estimate: The Shannon Number 11 [

12 An Estimate: The Shannon Number 12 [

13 An Estimate: The Shannon Number 13 [

14 The Shannon Number: Just an Estimate 14 [

15 15 Example: Sock It Two Me

In his haste to get ready for school, (and still a bit sleepy), he reaches into his drawer and pulls out 2 socks.

16 [Greenes, 1977] Example: Sock It Two Me 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? 16

Origin of Probability Theory Probability theory was originally inspired by gambling problems. [http://www.youtube.com/watch?

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

17 Origin of Probability Theory Probability theory was originally inspired by gambling problems. [ 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. 17 best known for Fermat's Last Theorem

a row. 18 Bet that them can t turn over three red cards.

18 Example: The Seven Card Hustle 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. 18 Bet that them can t turn over three red cards. The probability that they CAN do it is ! 3! 7 3! 2! 4! ! [Lovell, 2006]

19 Permutation 19 [

20 Permutation There are 11! = 39,916,800 ways to permute 11 persons. 20 [

21 Finger-Smudge on Touch-Screen Devices 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. [ 21

22 For sale Andre Woolery Art 22 [

23 Andre Woolery Art Fruit Ninja Facebook 23 Angry Bird Mail

24 Lockscreen PIN / Passcode 24 [

25 Smudge Attack Touchscreen smudge may give away your password/passcode Four distinct fingerprints reveals the four numbers used for passcode lock. 25 [

26 Suggestion: Repeat One Digit Unknown numbers: The number of 4-digit different passcodes = 10 4 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 = 2 50% improvement 26 Choose the number that will be repeated Choose the locations of the two nonrepeated numbers.

by only 10 different passcodes 27 [http://amitay.

27 News: Most Common Lockscreen PINs Passcodes of users of Big Brother Camera Security iphone app 15% of all passcode sets were represented by only 10 different passcodes 27 [ (2011)] out of 204,508 recorded passcodes

Even easier in Splinter Cell Decipher the

28 Even easier in Splinter Cell Decipher the keypad's code by the heat left on the buttons. Here's the keypad viewed with your thermal goggles. (Numbers added for emphasis.) Again, the stronger the signature, the more recent the keypress. The code is

Actual Research 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 imageanalysis software can

29 Actual Research 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 imageanalysis 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 Heat of the Moment: Characterizing the Efficacy of Thermal-Camera Based Attacks. Proceedings of WOOT

30 Alternative Solution Scramble PIN Layout in CyanogenMod (a modified version of Android ) 30 Scramble the keypad when you unlock your phone so that people can't peek at your keystrokes and learn your PIN.

31 The Birthday Problem (Paradox) 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? 31

32 The Birthday Problem (con t) 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 Mathematical Monthly, Vol. 73, No.4, 1966, pp ] 32

33 Birthday Coincidence: 2 nd Version 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? 33

34 Permuting four distinct letters: A,B,C,D DCBA CDBA BCDA ACBD DCAB CDAB BCAD ACDB DBCA CBDA BDCA ABCD DBAC CBAD BDAC ABDC DABC CABD BADC ADBC DACB CADB BACD ADCB 34

35 Permuting four letters: A,A,B,C D ACBA CABA BCAA ACBA ACAB CAAB BCAA ACAB ABCA CBAA BACA ABCA ABAC CBAA BAAC ABAC AABC CABA BAAC AABC AACB CAAB BACA AACB 35

36 Permuting four letters: A,A,B,C D ACBA CABA BCAA ACBA ACAB CAAB BCAA ACAB ABCA CBAA BACA ABCA ABAC CBAA BAAC ABAC AABC CABA BAAC AABC AACB CAAB BACA AACB 36

37 Permuting four letters: A,A,B,C D DCBA CDBA BCDA ACBD DCAB CDAB BCAD ACDB DBCA CBDA BDCA ABCD DBAC CBAD BDAC ABDC DABC CABD BADC ADBC DACB CADB BACD ADCB 37

38 Permuting four letters: A,A,B,C Group 1 D DCBA CDBA BCDA ACBD DCAB CDAB BCAD ACDB DBCA CBDA BDCA ABCD DBAC CBAD BDAC ABDC DABC CABD BADC ADBC DACB CADB BACD ADCB Group 2 38

39 Distinct Passcodes (revisit) Unknown numbers: The number of 4-digit different passcodes = 10 4 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: ,000

40 Need more practice? [ ] Ex: Poker Probability 40

41 [ Ex: Poker Probability 41

42 [ Ex: Poker Probability 42

43 [ Ex: Poker Probability 43

44 Binomial Theorem ( x y ) ( x y ) x1x2 x1y2 y1x2 yy 1 2 ( x y ) ( x y ) ( x y ) x1xx 2 3 xxy x1y 2x3 x1yy 2 3 yxx y1x2y3 y1y2x3 yyy x1 x2 x3 x y y y y ( x y) ( x y) xx xy yx yy x 2xy y ( x y) ( x y) ( x y) xxx xxy xyx xyy yxx yxy yyx yyy x xy 3 3xy y 2 2

45 How many chess games are possible?: A Revisit 53 [

Sequence A048987 0 1 1 20 2 400 3 8902 4 197281 (16 pawn moves and 4 knight moves) 5 4865609 6 119060324 7 3195901860 8 84998978956 9 2439530234167 10 69352859712417 11 2097651003696806 12

46 Sequence A (16 pawn moves and 4 knight moves) 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 [

47 It all began at Cornell 55 [

[http://www.ieee.org/documents/hamming_rl.pdf] IEEE Richard W. Hamming Medal 1988 - Richard W. Hamming 1989 - Irving S. Reed 1990 - Dennis M. Ritchie and Kenneth L. Thompson 1991 - Elwyn R.

48 [ IEEE Richard W. Hamming Medal Richard W. Hamming Irving S. Reed Dennis M. Ritchie and Kenneth L. Thompson Elwyn R. Berlekamp Lotfi A. Zadeh Jorma J. Rissanen Gottfried Ungerboeck Jacob Ziv Mark S. Pinsker Thomas M. Cover David D. Clark David A. Huffman Solomon W. Golomb A. G. Fraser Peter Elias Claude Berrou and Alain Glavieux Jack K. Wolf Neil J.A. Sloane Vladimir I. Levenshtein Abraham Lempel Sergio Verdú Peter Franaszek Whitfield Diffie, Martin Hellman and Ralph Merkle Toby Berger Michael Luby, Amin Shokrollahi Arthur Robert Calderbank Thomas Richardson and Rüdiger L. Urbanke 56 For contributions to coding theory and its applications to communications, computer science, mathematics and statistics.

Major contributions are in the fields of combinatorics, errorcorrecting codes, and sphere packing.

49 57 Neil J.A. Sloane 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 Major contributions are in the fields of combinatorics, errorcorrecting codes, and sphere packing. Best known for being the creator and maintainer of the On-Line Encyclopedia of Integer Sequences. [

50 How good was the Shannon s estimation? Actual number of possible games Shannon's estimates #moves

51 It is quite easy to get a better approximation Actual number of possible games Shannon's estimates Prapun's estimates #moves

Applications of Probability Theory

Applications of Probability Theory The subject of probability can be traced back to the 17th century when it arose out of the study of gambling games. The range of applications extends beyond games into