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History of Probability Theory

History of Probability Theory. Started in the year of 1654 De Mere (a well-known gambler) asked a question to Blaise Pascal (a mathematician). Whether to bet on the following event? “To throw a pair of dice 24 times, if a ‘double six’ occurs at least once, then win.”. correspond.

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History of Probability Theory

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  1. History of Probability Theory • Started in the year of 1654 • De Mere (a well-known gambler) asked a question to Blaise Pascal (a mathematician) • Whether to bet on the following event? • “To throw a pair of dice 24 times, if a ‘double six’ occurs at least once, then win.” correspond Blaise Pascal Pierre Fermat BUS304 – Probability Theory

  2. Applications of Probability Theory • Gambling: • Poker games, lotteries, etc. • Weather report: • Likelihood to rain today • Power of Katrina • Many more in modern business world • Risk Management and Investment • Value of stocks, options, corporate debt; • Insurance, credit assessment, loan default • Industrial application • Estimation of the life of a bulb, the shipping date, the daily production BUS304 – Probability Theory

  3. Example: Experiment Experimental Outcomes Toss a coin Head, tail Inspect a part Defective, nondefective Play a football game Win, lose, tie Roll a die Concept: Experiment and event • Experiment: A process that produces a single outcome whose result cannot be predicted with certainty. • Event: A certain outcome obtained in an experiment. Example of an event (description of outcome) • Two heads in a row when you flip a coin three times; • At least one “double six” when you throw a pair of dice 24 times. BUS304 – Probability Theory

  4. Description of Events • Elementary Events • The most rudimentary outcomes resulting from a simple experiment • Throwing one die, “obtaining a ” is an elementary event • Denoted as “e1, e2, …, en” Note: the elementary events cannot be further divided into smaller events. e.g. flip a coin twice, how many elementary events you expect to observe? • “getting one head one tail” is NOT an elementary event. • Elementary events are {HH, HT, TH, TT} BUS304 – Probability Theory

  5. Description of Events • Sample Space: • Collection of all elementary outcomes: • In many experiments, identifying sample space is important. • Write down the sample space of the following experiments: • throwing a pair of dice. • flipping a coin three times. • drawing two cards from a bridge deck. • An event (denoted as E), can be represented as a combination of elementary events. • E.g. E = A die shows number higher than 3 Elementary events: e1 = ; e2 = ; e3= . BUS304 – Probability Theory

  6. Rules of Assigning Probabilities • Three rules are commonly used: • Classical Probability Assessment • Relative Frequency Assessment • Subjective Probability Assessment BUS304 – Probability Theory

  7. Classical probability Assessment: Basic Rules to assign probability (1) Exercise: Decide the probability of the following events • Get a card higher than 10 from a bridge deck • Get a sum higher than 11 from throwing a pair of dice. • John and Mike both randomly pick a number from 1-5, what is the chance that these two numbers are the same? Number of Elementary Events Total number of Elementary Events P(E) = • where: • E refers to a certain event. • P(E) represents the probability of the event E When to use this rule? When the chance of each elementary event is the same: e.g. cards, coins, dices, use random number generator to select a sample BUS304 – Probability Theory

  8. Basic Rules to assign probability (2) • Relative Frequency of Occurrence Number of times E occurs N Probability of Future Event = Relative Freq. of Past = Examples: • If a survey result says, among 1000 people, 600 prefer iphone to ipod touch, then you assign the probability that the next person you meet will like iphone is 60%. • A basketball player’s percentage of made free throws. Why do you think Yao Ming has a better chance to win the free throw competition than Shaq O’Neal? • The probability that a TV is sent back for repair? Based on past experience. • The most commonly used in the business world. BUS304 – Probability Theory

  9. Exercise • A clerk recorded the number of patients waiting for service at 9:00am on 20 successive days Assign the probability that there are at most 2 agents waiting at 9:00am. BUS304 – Probability Theory

  10. Exercise 4.1 (Page 137) Elementary Events? Sample Space? a) Probability that “a customer is a male”? b) Probability that “a customer is 20 to 40 years old”? c) Probability that “a customer being 20 to 40 years old and a male”? BUS304 – Probability Theory

  11. Basic Rules to assign probability (3) • Subjective Probability Assessment • Subjective probability assessment has to be used when there is not enough information for past experience. • Example1: The probability a player will make the last minute shot (a complicated decision process, contingent on the decision by the component team’s coach, the player’s feeling, etc.) • Example2: Deciding the probability that you can get the job after the interview. • Smile of the interviewer • Whether you answer the question smoothly • Whether you show enough interest of the position • How many people you know are competing with you • Etc. • Always try to use as much information as possible. • As the world is changing dramatically, people are more and more rely upon subjective assessment. BUS304 – Probability Theory

  12. Summary of Basic Approaches • Classical Rule • Elementary events have equal odds • Relative Frequency • Use relative frequency table. Probability assigned based on percentage of occurrence. • Subjective • Based on experience, combining different signals to make inference. No standard approach to have people agree on each other. No matter what method used, probability cannot be higher than 1 or lower than 0! BUS304 – Probability Theory

  13. Rules for complement events • what is the a complement event? • The Rule: E E If Obama’s chance of winning the presidential campaign is assigned to be 60%, that means McCain’s chance is 1-60% = 40%. If the probability that at most two patients are waiting in the line is 0.65, what is the complement event? And what is the probability? BUS304 – Probability Theory

  14. Composite Events • E = E1 and E2 =(E1 is observed) AND (E2 is also observed) • E = E1 or E2 = Either (E1 is observed) Or (E2 is observed) More specifically, P(E1or E2) = P(E1) + P(E2) - P(E1and E2) E1 E2 P(E1and E2) ≤P(E1) P(E1and E2)≤P(E2) P(E1and E2) E1 or E2 E1 E2 P(E1or E2) ≥P(E1) P(E1or E2)≥P(E2) BUS304 – Probability Theory

  15. Exercise What is the probability of selecting a person who is a male? What is the probability of selecting a person who is under 20? What is the probability of selecting a person who is a male and also under 20? What is the probability of selecting a person who is either a male or under 20? BUS304 – Probability Theory

  16. Mutually Exclusive Events • If two events cannot happen simultaneously, then these two events are called mutually exclusive events. • Ways to determine whether two events are mutually exclusive: • If one happens, then the other cannot happen. Examples: • Draw a card, E1 = A Red card, E2 = A card of club • Throwing a pair of dice, E1 = one die shows E2 = a double six. • All elementary events are mutually exclusive. • Complement Events E2 E1 BUS304 – Probability Theory

  17. Rules for mutually exclusive events • If E1 and E2 are mutually exclusive, then • P(E1 and E2) = ? • P(E1 or E2) = ? • Exercise: • Throwing a pair of dice, what is the probability that I get a sum higher than 10? • E1: getting 11 • E2: getting 12 • E1 and E2 are mutually exclusive. • So P(E1 or E2) = P(E1) + P(E2) E2 E1 BUS304 – Probability Theory

  18. Conditional Probabilities • Information reveals gradually, your estimation changes as you know more. • Draw a card from bridge deck (52 cards). Probability of a spade card? • Now, I took a peek, the card is black, what is the probability of a spade card? • If I know the card is red, what is the probability of a spade card? • What is the probability of E1? • What if I know E2 happens, would you change your estimation? E1 E2 BUS304 – Probability Theory

  19. Conditional Probability Rule: Example: P(“Male”)=? P(“GPA 3.0”)=? P(“Male” and “GPA<3.0”)=? P(“Female” and “GPA 3.0”)=? P(“GPA<3.0” | “Male”) = ? P (“Female” | “GPA 3.0”)=? Bayes’ Theorem Thomas Bayes (1702-1761) BUS304 – Probability Theory

  20. Independent Events • If then we say that “Events E1 and E2 are independent”. That is, the outcome of E1 is not affected by whether E2 occurs. • Typical Example of independent Events: • Throwing a pair of dice, “the number showed on one die” and “the number on the other die”. • Toss a coin many times, the outcome of each time is independent to the other times. How to prove? 20

  21. Exercise • Calculate the following probabilities: • Prob of getting 3 heads in a row? • Prob of a “double-six”? • Prob of getting a spade card which is also higher than 10? • Data shown from the following table. Decide whether the following events are independent? • “Selecting a male” versus “selecting a female”? • “Selecting a male” versus “selecting a person under 20”? BUS304 – Probability Theory

  22. Probability Distribution • Random Variable: • A variable with random (unknown) value. • Examples • 1. Roll a die twice: Let x be the number of times 4 comes up. • x = 0, 1, or 2 • 2. Toss a coin 5 times: Let x be the number of heads • x = 0, 1, 2, 3, 4, or 5 • 3. Same as experiment 2: Let’s say you pay your friend $1 every time head shows up, and he pays you $1 otherwise. Let x be amount of money you gain from the game. • What are the possible values of x? BUS304 – Probability Theory

  23. Discrete vs. Continuous Random variables Random Variables Discrete Continuous • Examples: • Examples: BUS304 – Probability Theory

  24. .50 .25 0 1 2 x Discrete Probability Distribution Two ways to represent discrete probability distributions Table All the possible values of x Probability Graph BUS304 – Probability Theory

  25. Exercise • Describe the probability distribution of the random variables: • Draw a pair of dice, x is the random variable representing the sum of the total points. Step 1: Write down all the possible values in left column • Step 1.1: Write down the sample space Step 2: Write down the corresponding probabilities BUS304 – Probability Theory

  26. x P(x) -2 .25 0 .50 2 .25 Measures of Discrete Random Variables Example: What is your expected gain when you play the flip-coin game twice? • Expected value of a discrete distribution • An weighted average, taking into account the probability • The expected value of random variable x is denoted as E(x) E(x)= xi P(xi) E(x)= x1P(x1) +x2P(x2) + … + xnP(xn) E(x) = (-2) * 0.25 + 0 * 0.5 + 2 * 0.25 = 0 Your expected gain is 0! – a fair game. BUS304 – Probability Theory

  27. Spreadsheet to compute the expected value • Step1: develop the distribution table according to the description of the problem. • Step2: add one (3rd) column to compute the product of the value and the probability • Step3: compute the sum of the 3rd column  The Expected Value E(x) =-0.5+0+0.5=0 BUS304 – Probability Theory

  28. Exercise • You are working part time in a restaurant. The amount of tip you get each time varies. Your estimation of the probability is as follows: • You bargain with the boss saying you want a more fixed income. He said he can give you $62 per night, instead of letting you keep the tips. Would you want to accept this offer? BUS304 – Probability Theory

  29. More Exercise • Buy lottery: price $10 • With 0.0000001 chance, you can win $1million • With 0.001 chance, you can win $1000 • With 0.1 chance, you can win $50 What is the expected gain of buying this lottery ticket? Is buying lottery a fair game?

  30. Rule for expected value • If there are two random variables, x and y. Then E(x+y) = E(x) + E(y) • Example: “Head -$2”, “Tail +1” • x is your gain from playing the game the first time • y is your gain from playing the game the second time • x+y is your total gain from playing the two games. Write down the probability distribution of x+y and calculate the expected value for x+y E(y)= -0.5 E(x)= -0.5 Is this game a fair game? BUS304 – Probability Theory

  31. Exercise • Assume that the expected payoff of playing the slot machine is -0.04 cents • What is the expected payoff when playing 100 times? 10,000 times?

  32. Measure of risk-- variance • Two games • Flip a coin once, if head then you get $1, otherwise you pay $1; • Flip a coin once, if head then you get $100, otherwise you pay $100; • Which game will you choose? • Three basic types of people • Risk-lover • Risk-neutral • Risk-averse What is your type?

  33. Measures – variance Step 1: develop the probability distribution table. Step 2: compute the mean E(x): 50x0.2+60x0.3+70x0.4+80x0.1=64 Step 3: compute the distance from the mean for each value (x-E(x)) Step 4: square each distance (x – E(x))2 Step 5: weight the squared distance: (x-E(x))2P(x) Step 6: sum up all the weighted square distance  variance • Variance: a weighted average of the squared deviation from the expected value. BUS304 – Probability Theory

  34. Variance The variance of a random variable has the same meaning as the variance of population Calculation is the same as calculating population variance using a relative frequency table. Written as var(x) or Standard deviation of a random variable: Same of the population standard deviation Calculate the variance Then take the square root of the variance. Written as sd(x) or e.g. for the example on page 10 Variance and Standard deviation BUS304 – Probability Theory

  35. More exercise: • Page 4.66 BUS304 – Probability Theory

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