History of Probability Theory Started in the year of 1654 a well-known gambler, De Mere asked a question to Blaise Pascal 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

Applications of Probability Theory The World is full of uncertainty! Knowing probability theory is important ! Gambling: Poker games, lotteries, etc. Weather report: Likelihood to rain today Power of Katrina Statistical Inferential 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 Reading information from the chart. BUS304 – Probability Theory

Concept: Experiment and event Experiment: A process of obtaining well-defined outcomes for uncertain events Event: A certain outcome in an experiment Example: Roll a die Win, lose, tie Play a football game Defective, nondefective Inspect a part Head, tail Toss a coin Experimental Outcomes Experiment Example: 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

Basic Rules to assign probability (1) Classical probability Assessment: Basic Rules to assign probability (1) where: E refers to a certain event. P(E) represent the probability of the event E 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? P(E) = Number of ways E can occur Total number of ways When to use this rule? When the chance of each way is the same: e.g. cards, coins, dices, use random number generator to select a sample BUS304 – Probability Theory

Basic Rules to assign probability (2) Find the relative frequency => probability Relative Frequency of Occurrence Relative Freq. of Ei = Number of times E occurs N Examples: If a survey result says, among 1000 people, 500 of them think the new 2GB ipod nano is much better than the 20GB ipod. Then you assign the probability that a person like Nano better is 50%. A basketball player’s proportion of made free throws The probability that a TV is sent back for repair The most commonly used in the business world. Reading information from the chart. BUS304 – Probability Theory

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. Number of waiting Number of Days Outcome Occurs 2 1 5 6 3 4 Total 20 BUS304 – Probability Theory

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. 6.31 days 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

Rules for complement events what is the a complement event? The Rule: E E If Bush’s chance of winning is assigned to be 60% before the election, that means Kerry’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

More Exercise (homework) Page 137 Problem 4.2 (a) (b) (c) Problem 4.5 Problem 4.8 (a) Problem 4.10 BUS304 – Probability Theory

BUS304 – Probability Theory 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(E1 or E2) = P(E1) + P(E2) - P(E1 and E2) E1 E2 P(E1 and E2) ≤ P(E1) P(E1 and E2) ≤ P(E2) P(E1 and E2) E1 or E2 E1 E2 P(E1 or E2) ≥ P(E1) P(E1 or E2) ≥ P(E2) BUS304 – Probability Theory

BUS304 – Probability Theory Exercise Male Female Total Under 20 168 208 376 20 to 40 340 290 630 Over 40 170 160 330 678 658 1336 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

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 E1 E2 BUS304 – Probability Theory

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) E1 E2 BUS304 – Probability Theory

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

BUS304 – Probability Theory Bayes’ Theorem 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”)=? Thomas Bayes (1702-1761) GPA3.0 GPA<3.0 Male 282 323 Female 305 318 BUS304 – Probability Theory

BUS304 – Probability Theory 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? BUS304 – Probability Theory 16

BUS304 – Probability Theory 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”? Male Female Under 20 168 208 20 to 40 340 290 Over 40 170 160 BUS304 – Probability Theory