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Asst. Prof. Dr. Prapun [email protected]
4 Combinatorics
1
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
2
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
3
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
4
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
8
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
11 [https://www.youtube.com/watch?v=Km024eldY1A]
An Estimate: The Shannon Number
12 [https://www.youtube.com/watch?v=Km024eldY1A]
An Estimate: The Shannon Number
13 [https://www.youtube.com/watch?v=Km024eldY1A]
The Shannon Number: Just an Estimate
14 [https://www.youtube.com/watch?v=Km024eldY1A]
Example: Sock It Two Me
15
Example: Sock It Two Me
16
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
17
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
18
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
27
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
20
There are 11! = 39,916,800 ways to permute 11 persons.
[https://www.youtube.com/watch?v=2EN6jYMiGF4]
Finger-Smudge on Touch-Screen Devices
21
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
25
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
26
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
27
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
28
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
29
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
30
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)
31
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)
32
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
33
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
34
DCBA
DCABDBCADBACDABC
DACB
CDBA
CDABCBDACBADCABD
CADB
BCDA
BCADBDCABDACBADC
BACD
ACBD
ACDBABCDABDCADBC
ADCB
Permuting four letters: A,A,B,C
35
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
36
Permuting four letters: A,A,B,CD
37
DCBA
DCABDBCADBACDABC
DACB
CDBA
CDABCBDACBADCABD
CADB
BCDA
BCADBDCABDACBADC
BACD
ACBD
ACDBABCDABDCADBC
ADCB
Permuting four letters: A,A,B,CD
38
DCBA
DCABDBCADBACDABC
DACB
CDBA
CDABCBDACBADCABD
CADB
BCDA
BCADBDCABDACBADC
BACD
ACBD
ACDBABCDABDCADBC
ADCB
Permuting four letters: A,A,B,CD
Group 1
Group 2
Distinct Passcodes (revisit)
39
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
40
[ http://en.wikipedia.org/wiki/Poker_probability ]Need more practice?
Ex: Poker Probability
41
[http://www.wolframalpha.com/input/?i=probability+of+royal+flush]
Ex: Poker Probability
42
[http://www.wolframalpha.com/input/?i=probability+of+full+house]
Ex: Poker Probability
43
[http://www.wolframalpha.com/input/?i=probability+3+queens+2+jacks&lk=3]
Binomial Theorem
44
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
53 [https://www.youtube.com/watch?v=Km024eldY1A]
Sequence A048987
54
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
56
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
57
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.
[htt
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How good was the Shannon’s estimation?
58
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