1、统计学基础(英文版第7版)教学课件les7e_ppt_04_02-(1)Chapter Outline 4.1 Probability Distributions 4.2 Binomial Distributions 4.3 More Discrete Probability Distributions.Section 4.2Binomial Distributions.Section 4.2 Objectives How to determine whether a probability experiment is a binomial experiment How to find bin
2、omial probabilities using the binomial probability formula How to find binomial probabilities using technology,formulas,and a binomial probability table How to construct and graph a binomial distribution How to find the mean,variance,and standard deviation of a binomial probability distribution.Bino
3、mial Experiments1.The experiment is repeated for a fixed number of trials,where each trial is independent of other trials.2.There are only two possible outcomes of interest for each trial.The outcomes can be classified as a success(S)or as a failure(F).3.The probability of a success,P(S),is the same
4、 for each trial.4.The random variable x counts the number of successful trials.Notation for Binomial ExperimentsSymbolDescriptionnThe number of times a trial is repeatedpThe probability of success in a single trialqThe probability of failure in a single trial(q=1 p)xThe random variable represents a
5、count of the number of successes in n trials:x=0,1,2,3,n.Example:Identifying and Understanding Binomial ExperimentsDecide whether each experiment is a binomial experiment.If it is,specify the values of n,p,and q,and list the possible values of the random variable x.If it is not,explain why.1.A certa
6、in surgical procedure has an 85%chance of success.A doctor performs the procedure on eight patients.The random variable represents the number of successful surgeries.Solution:Identifying and Understanding Binomial ExperimentsBinomial Experiment1.Each surgery represents a trial.There are eight surger
7、ies,and each one is independent of the others.2.There are only two possible outcomes of interest for each surgery:a success(S)or a failure(F).3.The probability of a success,P(S),is 0.85 for each surgery.4.The random variable x counts the number of successful surgeries.Solution:Identifying and Unders
8、tanding Binomial ExperimentsBinomial Experiment n=8(number of trials)p=0.85(probability of success)q=1 p=1 0.85=0.15(probability of failure)x=0,1,2,3,4,5,6,7,8(number of successful surgeries).Example:Identifying and Understanding Binomial ExperimentsDecide whether each experiment is a binomial exper
9、iment.If it is,specify the values of n,p,and q,and list the possible values of the random variable x.If it is not,explain why.2.A jar contains five red marbles,nine blue marbles,and six green marbles.You randomly select three marbles from the jar,without replacement.The random variable represents th
10、e number of red marbles.Solution:Identifying and Understanding Binomial ExperimentsNot a Binomial Experiment The probability of selecting a red marble on the first trial is 5/20.Because the marble is not replaced,the probability of success(red)for subsequent trials is no longer 5/20.The trials are n
11、ot independent and the probability of a success is not the same for each trial.Binomial Probability FormulaBinomial Probability Formula The probability of exactly x successes in n trials is!()()!xn xxn xnxnP xC p qp qnxxn=number of trialsp=probability of successq=1 p probability of failurex=number o
12、f successes in n trialsNote:number of failures is n x.Example:Finding a Binomial ProbabilityRotator cuff surgery has a 90%chance of success.The surgery is performed on three patients.Find the probability of the surgery being successful on exactly two patients.(Source:The Orthopedic Center of St.Loui
13、s).Solution:Finding a Binomial ProbabilityMethod 1:Draw a tree diagram and use the Multiplication Rule.81310000.243 Solution:Finding a Binomial ProbabilityMethod 2:Use the binomial probability formula.213!91(2)(32)!2!101081131001081310000.243P Binomial Probability DistributionBinomial Probability Di
14、stribution List the possible values of x with the corresponding probability of each.Example:Binomial probability distribution for Microfacture knee surgery:n=3,p=Use binomial probability formula to find probabilities.x0123P(x)0.0160.1410.4220.42234.Example:Constructing a Binomial DistributionIn a su
15、rvey,U.S.adults were asked to identify which social media platforms they use.The results are shown in the figure.Six adults who participated in the survey are randomly selected and asked whether they use the social media platform Facebook.Construct a binomial probability distribution for the number
16、of adults who respond yes.(Source:Pew Research).Solution:Constructing a Binomial Distributionp=0.68 and q=0.32n=6,possible values for x are 0,1,2,3,4,5 and 6.060660(0)0.680.321 0.680.320.001PC 151561(1)0.680.326 0.680.320.014PC 242462(2)0.680.3215 0.680.320.073PC 333363(3)0.680.3220 0.680.320.206PC
17、424264(4)0.680.3215 0.680.320.328PC 515165(5)0.680.326 0.680.320.279PC 606066(6)0.680.321 0.680.320.099PCSolution:Constructing a Binomial DistributionNotice in the table that all the probabilities are between 0 and 1 and that the sum of the probabilities is 1.Example:Finding a Binomial Probabilities
18、 Using TechnologyA survey found that 26%of U.S.adults believe there is no difference between secured and unsecured wireless networks.(A secured network uses barriers,such as firewalls and passwords,to protect information;an unsecured network does not.)You randomly select 100 adults.What is the proba
19、bility that exactly 35 adults believe there is no difference between secured and unsecured networks?Use technology to find the probability.(Source:University of Phoenix).Solution:Finding a Binomial Probabilities Using Technology.SolutionMinitab,Excel,StatCrunch,and the TI-84 Plus each have features
20、that allow you to find binomial probabilities.Try using these technologies.You should obtain results similar to these displays.Solution:Finding a Binomial Probabilities Using Technology.SolutionFrom these displays,you can see that the probability that exactly 35 adults believe there is no difference
21、 between secured and unsecured networks is about 0.012.Because 0.012 is less than 0.05,this can be considered an unusual event.Example:Finding Binomial Probabilities Using FormulasA survey found that 17%of U.S.adults say that Google News is a major source of news for them.You randomly select four ad
22、ults and ask them whether Google News is a major source of news for them.Find the probability that(1)exactly two of them respond yes,(2)at least two of them respond yes,and(3)fewer than two of them respond yes.(Source:Ipsos Public Affairs).Solution:Finding Binomial Probabilities Using Formulas.Solut
23、ion:Finding Binomial Probabilities Using FormulasSolution2.To find the probability that at least two adults will respond yes,find the sum of P(2),P(3),and P(4).Begin by using the binomial probability formula to write an expression for each probability.P(2)=4C2(0.17)2(0.83)2=6(0.17)2(0.83)2P(3)=4C3(0
24、.17)3(0.83)1=4(0.17)3(0.83)1P(4)=4C4(0.17)4(0.83)0=1(0.17)4(0.83)0.Solution:Finding Binomial Probabilities Using Formulas.Solution:Finding Binomial Probabilities Using Formulas.Example:Finding a Binomial Probability Using a TableAbout 10%of workers(ages 16 years and older)in the United States commut
25、e to their jobs by carpooling.You randomly select eight workers.What is the probability that exactly four of them carpool to work?Use a table to find the probability.(Source:American Community Survey)Solution:Binomial with n=8,p=0.1,x=4.Solution:Finding Binomial Probabilities Using a Table A portion
26、 of Table 2 is shownAccording to the table,the probability is 0.005.Solution:Finding Binomial Probabilities Using a Table You can check the result using technology.So,the probability that exactly four of the eight workers carpool to work is 0.005.Because 0.005 is less than 0.05,this can be considere
27、d an unusual event.Example:Graphing a Binomial DistributionSixty-two percent of cancer survivors are ages 65 years or older.You randomly select six cancer survivors and ask them whether they are 65 years of age or older.Construct a probability distribution for the random variable x.Then graph the di
28、stribution.(Source:National Cancer Institute)Solution:n=6,p=0.62,q=0.38 Find the probability for each value of x.Solution:Graphing a Binomial Distribution.Notice in the table that all the probabilities are between 0 and 1 and that the sum of the probabilities is 1.Solution:Graphing a Binomial Distri
29、butionHistogram:.From the histogram,you can see that it would be unusual for none or only one of the survivors to be age 65 years or older because both probabilities are less than 0.05.Mean,Variance,and Standard Deviation Mean:=np Variance:2=npq Standard Deviation:npq.Example:Mean,Variance,and Stand
30、ard DeviationIn Pittsburgh,Pennsylvania,about 56%of the days in a year are cloudy.Find the mean,variance,and standard deviation for the number of cloudy days during the month of June.Interpret the results and determine any unusual values.(Source:National Climatic Data Center)Solution:n=30,p=0.56,q=0
31、.44Mean:=np=300.56=16.8Variance:2=npq=300.560.44 7.4Standard Deviation:30 0.56 0.442.7npq.Solution:Mean,Variance,and Standard Deviation=16.8 2 7.4 2.7 On average,there are 16.8 cloudy days during the month of June.The standard deviation is about 2.7 days.Values that are more than two standard deviations from the mean are considered unusual.16.8 2(2.7)=11.4;A June with 11 cloudy days or less would be unusual.16.8+2(2.7)=22.2;A June with 23 cloudy days or more would also be unusual.
