History of Probability TheoryApplications of Probability TheoryConcept: Experiment and eventDescription of EventsSlide 5Rules of Assigning ProbabilitiesBasic Rules to assign probability (1)Basic Rules to assign probability (2)ExerciseExercise 4.1 (Page 137)Basic Rules to assign probability (3)Summary of Basic ApproachesRules for complement eventsComposite EventsSlide 15Mutually Exclusive EventsRules for mutually exclusive eventsConditional ProbabilitiesBayes’ TheoremIndependent EventsSlide 21Probability DistributionDiscrete vs. Continuous Random variablesDiscrete Probability DistributionSlide 25Measures of Discrete Random VariablesSpreadsheet to compute the expected valueSlide 28More ExerciseRule for expected valueSlide 31Measure of risk-- varianceMeasures – varianceVariance and Standard deviationMore exercise:01/14/19 BUS304 – Probability Theory 1History 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.”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.”correspondBlaise Pascal Pierre Fermat01/14/19 BUS304 – Probability Theory 2Applications 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 production01/14/19 BUS304 – Probability Theory 3Concept: 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.Example:Roll a dieWin, lose, tiePlay a football gameDefective, nondefectiveInspect a partHead, tailToss a coinExperimental OutcomesExperimentDescription 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}01/14/19 BUS304 – Probability Theory 4Description 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 3Elementary events: e1 = ; e2 = ; e3= .01/14/19 BUS304 – Probability Theory 5Rules of Assigning ProbabilitiesThree rules are commonly used:Classical Probability AssessmentRelative Frequency AssessmentSubjective Probability Assessment01/14/19 BUS304 – Probability Theory 601/14/19 BUS304 – Probability Theory 7Basic Rules to assign probability (1)P(E) =Number of Elementary EventsTotal number of Elementary EventsClassical probability Assessment: where:• E refers to a certain event. •P(E) represents the probability of the event EWhen 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 sampleExercise: Decide the probability of the following events1. Get a card higher than 10 from a bridge deck2. Get a sum higher than 11 from throwing a pair of dice.3. John and Mike both randomly pick a number from 1-5, what is the chance that these two numbers are the same?01/14/19 BUS304 – Probability Theory 8Basic Rules to assign probability (2)Relative Frequency of OccurrenceExamples: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.Probability of Future Event = Relative Freq. of Past =Number of times E occursN01/14/19 BUS304 – Probability Theory 9ExerciseA clerk recorded the number of patients waiting for service at 9:00am on 20 successive daysNumber of waiting Number of Days Outcome Occurs0 21 52 63 4≥ 4 3 Total 20Assign the probability that there are at most 2 agents waiting at 9:00am.Exercise 4.1 (Page 137)Male FemaleUnder 20 168 20820 to 40 340 290Over 40 170 16001/14/19 BUS304 – Probability Theory 10Elementary 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”?01/14/19 BUS304 – Probability Theory 11Basic 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.Summary of Basic ApproachesClassical RuleElementary events have equal oddsRelative FrequencyUse relative frequency table. Probability assigned based on