统计学第五章.ppt

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1、The McGraw-Hill Companies,Inc.2008McGraw-Hill/IrwinA Survey of Probability ConceptsChapter 5GOALSlDefine probability.lDescribe the classical,empirical,and subjective approaches to probability.lExplain the terms experiment,event,outcome,permutations,and combinations.lDefine the terms conditional prob

2、ability and joint probability.lCalculate probabilities using the rules of addition and rules of multiplication.lApply a tree diagram to organize and compute probabilities.lCalculate a probability using Bayes theorem.2DefinitionsA probability is a measure of the likelihood that an event in the future

3、 will happen.It it can only assume a value between 0 and 1.lA value near zero means the event is not likely to happen.A value near one means it is likely.lThere are three ways of assigning probability:classical,empirical,and subjective.3Probability Examples4Definitions continuedlAn experiment is the

4、 observation of some activity or the act of taking some measurement.lAn outcome is the particular result of an experiment.lAn event is the collection of one or more outcomes of an experiment.5Experiments,Events and Outcomes6Assigning ProbabilitiesThree approaches to assigning probabilitiesClassicalE

5、mpirical Subjective7Classical ProbabilityConsider an experiment of rolling a six-sided die.What is the probability of the event“an even number of spots appear face up”?The possible outcomes are:There are three“favorable”outcomes(a two,a four,and a six)in the collection of six equally likely possible

6、 outcomes.8Mutually Exclusive EventslEvents are mutually exclusive if the occurrence of any one event means that none of the others can occur at the same time.lEvents are independent if the occurrence of one event does not affect the occurrence of another.9Collectively Exhaustive EventslEvents are c

7、ollectively exhaustive if at least one of the events must occur when an experiment is conducted.10Empirical ProbabilityThe empirical approach to probability is based on what is called the law of large numbers.The key to establishing probabilities empirically is that more observations will provide a

8、more accurate estimate of the probability.11Law of Large NumbersSuppose we toss a fair coin.The result of each toss is either a head or a tail.If we toss the coin a great number of times,the probability of the outcome of heads will approach.5.The following table reports the results of an experiment

9、of flipping a fair coin 1,10,50,100,500,1,000 and 10,000 times and then computing the relative frequency of heads12Empirical Probability-ExampleOn February 1,2003,the Space Shuttle Columbia exploded.This was the second disaster in 113 space missions for NASA.On the basis of this information,what is

10、the probability that a future mission is successfully completed?13Subjective Probability-ExamplelIf there is little or no past experience or information on which to base a probability,it may be arrived at subjectively.lIllustrations of subjective probability are:1.Estimating the likelihood the New E

11、ngland Patriots will play in the Super Bowl next year.2.Estimating the likelihood you will be married before the age of 30.3.Estimating the likelihood the U.S.budget deficit will be reduced by half in the next 10 years.14Summary of Types of Probability15Rules for Computing ProbabilitiesRules of Addi

12、tionlSpecial Rule of Addition-If two events A and B are mutually exclusive,the probability of one or the other events occurring equals the sum of their probabilities.P(A or B)=P(A)+P(B)lThe General Rule of Addition-If A and B are two events that are not mutually exclusive,then P(A or B)is given by t

13、he following formula:P(A or B)=P(A)+P(B)-P(A and B)16Addition Rule-ExampleWhat is the probability that a card chosen at random from a standard deck of cards will be either a king or a heart?P(A or B)=P(A)+P(B)-P(A and B)=4/52 +13/52-1/52=16/52,or.307717The Complement RuleThe complement rule is used

14、to determine the probability of an event occurring by subtracting the probability of the event not occurring from 1.P(A)+P(A)=1 or P(A)=1-P(A).18Joint Probability Venn DiagramJOINT PROBABILITY A probability that measures the likelihood two or more events will happen concurrently.19Special Rule of Mu

15、ltiplicationlThe special rule of multiplication requires that two events A and B are independent.lTwo events A and B are independent if the occurrence of one has no effect on the probability of the occurrence of the other.lThis rule is written:P(A and B)=P(A)P(B)20Multiplication Rule-ExampleA survey

16、 by the American Automobile association(AAA)revealed 60 percent of its members made airline reservations last year.Two members are selected at random.What is the probability both made airline reservations last year?Solution:The probability the first member made an airline reservation last year is.60

17、,written as P(R1)=.60The probability that the second member selected made a reservation is also.60,so P(R2)=.60.Since the number of AAA members is very large,you may assume thatR1 and R2 are independent.P(R1 and R2)=P(R1)P(R2)=(.60)(.60)=.3621Conditional ProbabilityA conditional probability is the p

18、robability of a particular event occurring,given that another event has occurred.The probability of the event A given that the event B has occurred is written P(A|B).22General Multiplication RuleThe general rule of multiplication is used to find the joint probability that two events will occur.Use t

19、he general rule of multiplication to find the joint probability of two events when the events are not independent.It states that for two events,A and B,the joint probability that both events will happen is found by multiplying the probability that event A will happen by the conditional probability o

20、f event B occurring given that A has occurred.23General Multiplication Rule-ExampleA golfer has 12 golf shirts in his closet.Suppose 9 of these shirts are white and the others blue.He gets dressed in the dark,so he just grabs a shirt and puts it on.He plays golf two days in a row and does not do lau

21、ndry.What is the likelihood both shirts selected are white?24lThe event that the first shirt selected is white is W1.The probability is P(W1)=9/12 lThe event that the second shirt selected is also white is identified as W2.The conditional probability that the second shirt selected is white,given tha

22、t the first shirt selected is also white,is P(W2|W1)=8/11.lTo determine the probability of 2 white shirts being selected we use formula:P(AB)=P(A)P(B|A)lP(W1 and W2)=P(W1)P(W2|W1)=(9/12)(8/11)=0.55General Multiplication Rule-Example25Contingency TablesA CONTINGENCY TABLE is a table used to classify

23、sample observations according to two or more identifiable characteristicsE.g.A survey of 150 adults classified each as to gender and the number of movies attended last month.Each respondent is classified according to two criteriathe number of movies attended and gender.26Contingency Tables-ExampleA

24、sample of executives were surveyed about their loyalty to their company.One of the questions was,“If you were given an offer by another company equal to or slightly better than your present position,would you remain with the company or take the other position?”The responses of the 200 executives in

25、the survey were cross-classified with their length of service with the company.What is the probability of randomly selecting an executive who is loyal to the company(would remain)and who has more than 10 years of service?27Event A1 happens if a randomly selected executive will remain with the compan

26、y despite an equal or slightly better offer from another company.Since there are 120 executives out of the 200 in the survey who would remain with the companyP(A1)=120/200,or.60.Event B4 happens if a randomly selected executive has more than 10 years of service with the company.Thus,P(B4|A1)is the c

27、onditional probability that an executive with more than 10 years of service would remain with the company.Of the 120 executives who would remain 75 have more than 10 years of service,so P(B4|A1)=75/120.Contingency Tables-Example28Tree DiagramsA tree diagram is useful for portraying conditional and j

28、oint probabilities.It is particularly useful for analyzing business decisions involving several stages.A tree diagram is a graph that is helpful in organizing calculations that involve several stages.Each segment in the tree is one stage of the problem.The branches of a tree diagram are weighted by

29、probabilities.2930Bayes TheoremlBayes Theorem is a method for revising a probability given additional information.lIt is computed using the following formula:31Bayes Theorem-Example32Bayes Theorem Example(cont.)33Bayes Theorem Example(cont.)34Bayes Theorem Example(cont.)3536Bayes Theorem Example(con

30、t.)37Counting Rules MultiplicationThe multiplication formula indicates that if there are m ways of doing one thing and n ways of doing another thing,there are m x n ways of doing both.Example:Dr.Delong has 10 shirts and 8 ties.How many shirt and tie outfits does he have?(10)(8)=80 38An automobile de

31、aler wants to advertise that for$29,999 you can buy a convertible,a two-door sedan,or a four-door model with your choice of either wire wheel covers or solid wheel covers.How many different arrangements of models and wheel covers can the dealer offer?Counting Rules Multiplication:Example39Counting R

32、ules Multiplication:Example40Counting Rules-PermutationA permutation is any arrangement of r objects selected from n possible objects.The order of arrangement is important in permutations.41Counting-CombinationA combination is the number of ways to choose r objects from a group of n objects without

33、regard to order.42Combination-ExampleThere are 12 players on the Carolina Forest High School basketball team.Coach Thompson must pick five players among the twelve on the team to comprise the starting lineup.How many different groups are possible?43Permutation-ExampleSuppose that in addition to selecting the group,he must also rank each of the players in that starting lineup according to their ability.44End of Chapter 545