Applications of Probability Theory

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1 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 business decisions, insurance, law, medical tests, and the social sciences. The stock market, the largest casino in the world, cannot do without it. (More in ECS 455) The telephone network, call centers, and airline companies with their randomly fluctuating loads could not have been economically designed without probability theory. 1

2 Probability and Random Processes ECS 315 Dr. Prapun Suksompong Set Theory 2 Office Hours: BKD Monday 14:40-16:00 Friday 14:00-16:00

3 3 Venn diagram

4 4 Venn diagram: Examples

5 5 Partition

6 Probability and Random Processes ECS 315 Dr. Prapun Suksompong Classical Probability 6 Office Hours: BKD Monday 14:40-16:00 Friday 14:00-16:00

7 Example In drawing a card from a deck, there are 52 equally likely outcomes, 13 of which are diamonds. This leads to a probability of 13/52 or 1/4. 7

8 The word dice Historically, dice is the plural of die. In modern standard English, dice is used as both the singular and the plural. 8 Example of 19th Century bone dice

9 Advanced dice 9 [ ]

10 Two dice: Simulation [ ] 10

11 Two dice Assume that the two dice are fair and independent. P[sum of the two dice = 5] = 4/36 11

12 Two dice Assume that the two dice are fair and independent. 12

13 Scandal of Arithmetic Which is more likely, obtaining at least one six in 4 tosses of a fair die (event A), or obtaining at least one double six in 24 tosses of a pair of dice (event B)? [ 13

14 Scandal of Arithmetic Which is more likely, obtaining at least one six in 4 tosses of a fair die (event A), or obtaining at least one double six in 24 tosses of a pair of dice (event B)? 5 PA ( ) PB ( )

15 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 on the previous slide. 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. 15 best known for Fermat's Last Theorem

16 Probability and Random Processes ECS 315 Dr. Prapun Suksompong Combinatorics 16 Office Hours: BKD Monday 14:40-16:00 Friday 14:00-16:00

17 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. [ 17

18 Lockscreen PIN / Passcode 18 [

19 Smudge Attack Touchscreen smudge may give away your password/passcode Four distinct fingerprints reveals the four numbers used for passcode lock. 19 [

20 Suggestion: Repeat One Digit Unknown numbers: The number of 4-digit different passcodes = 10 4 Exactly four different numbers: The number of 4-digit different passcodes = 24 Exactly three different numbers: The number of 4-digit different passcodes = 36 20

21 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? 21

22 Probability of birthday coincidence Probability that there is at least two people who have the same birthday in a group of r persons 1, if r r 1 1, r terms if 0 r

23 23 Probability of birthday coincidence

24 Zoom in

25 Triangular number A triangular number or triangle number is the number of dots in an equilateral triangle uniformly filled with dots. T ( n 1) n n nn ( 1) n Proof w/o Words 25

26 26 Sums of Squares

27 Sums of Cubes The sum of the first n cubes is the square of the nth triangular number. Visual Proof: n i ( i) n 3 2 i 1 i 1 n 3 = n copies of an n-by-n square 27

28 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 28

29 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? 29

30 Binomial Theorem ( x y ) ( x y ) x1x 2 x1 y2 y1x2 y1 y2 ( x y ) ( x y ) ( x y ) x1x 2x3 x1x 2y3 x1 y2x3 x1 y2y3 y1x2 x3 y1x2 y3 y1 y2x3 y1y2 y3 x x x 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 3xy 3xy y 2 2

31 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 = 2 Exactly two different numbers: The number of 4-digit different passcodes = Exactly one number: The number of 4-digit different passcodes = 1 Check:

32 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. 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]

33 [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? 33

34 Example: Sock It Two Me (2) Jack knows that 4 blue socks, 3 green socks, and 2 tan socks are in his drawer. He reaches into his drawer and pulls out 2 socks. 1. What are Jack's chances that he pulls out 2 blue socks to match his blue slacks? What are the chances that he pulls out a pair of matching socks?

35 Need more practice? Ex: Poker Probability [ ] 35

36 Watch Mlodinow s talk Delivered to Google employees About his book ( The Drunkard's Walk ) 36

37 Exercise At 10:14 into the video, Mlodinow shows three probabilities. Can you derive the first two? [Mlodinow, 2008, p ] 37

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