1、1Copyright 2019,2015,2012,Pearson Education,Inc.ChapterDiscrete Probability Distributions42Copyright 2019,2015,2012,Pearson Education,Inc.Chapter Outline 4.1 Probability Distributions 4.2 Binomial Distributions 4.3 More Discrete Probability Distributions.3Copyright 2019,2015,2012,Pearson Education,I
2、nc.Section 4.2Binomial Distributions.4Copyright 2019,2015,2012,Pearson Education,Inc.Section 4.2 Objectives How to determine whether a probability experiment is a binomial experiment How to find binomial probabilities using the binomial probability formula How to find binomial probabilities using te
3、chnology,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.5Copyright 2019,2015,2012,Pearson Education,Inc.Binomial Experiments1.The experiment is repeated for a fix
4、ed 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 for each trial.4.The random variable x counts the num
5、ber of successful trials.6Copyright 2019,2015,2012,Pearson Education,Inc.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 count
6、of the number of successes in n trials:x=0,1,2,3,n.7Copyright 2019,2015,2012,Pearson Education,Inc.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 var
7、iable x.If it is not,explain why.1.A certain 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.8Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Identifying and Understanding Binom
8、ial ExperimentsBinomial Experiment1.Each surgery represents a trial.There are eight surgeries,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 r
9、andom variable x counts the number of successful surgeries.9Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Identifying and Understanding Binomial ExperimentsBinomial Experimentn=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(num
10、ber of successful surgeries).10Copyright 2019,2015,2012,Pearson Education,Inc.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,
11、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 the number of red marbles.11Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Identifying and Understanding Bi
12、nomial ExperimentsNot a Binomial ExperimentThe 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 not independent and the probability of a success is not the same fo
13、r each trial.12Copyright 2019,2015,2012,Pearson Education,Inc.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 of successes in n
14、trialsNote:number of failures is n x.13Copyright 2019,2015,2012,Pearson Education,Inc.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:
15、The Orthopedic Center of St.Louis).14Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Finding a Binomial ProbabilityMethod 1:Draw a tree diagram and use the Multiplication Rule.81310000.243 15Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Finding a Binomial ProbabilityMethod 2:Use th
16、e binomial probability formula.213!91(2)(32)!2!101081131001081310000.243P 16Copyright 2019,2015,2012,Pearson Education,Inc.Binomial Probability DistributionBinomial Probability Distribution List the possible values of x with the corresponding probability of each.Example:Binomial probability distribu
17、tion for Microfacture knee surgery:n=3,p=Use binomial probability formula to find probabilities.x0123P(x)0.0160.1410.4220.42234.17Copyright 2019,2015,2012,Pearson Education,Inc.Example:Constructing a Binomial DistributionIn a survey,U.S.adults were asked to identify which social media platforms they
18、 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 of adults who respond yes.(Source:Pew Research).18Copyright 2019,2015,20
19、12,Pearson Education,Inc.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 424264(4)0.680.3215 0.6
20、80.320.328PC 515165(5)0.680.326 0.680.320.279PC 606066(6)0.680.321 0.680.320.099PC19Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Constructing a Binomial DistributionNotice in the table that all the probabilities are between 0 and 1 and that the sum of the probabilities is 1.20Copyright 20
21、19,2015,2012,Pearson Education,Inc.Example:Finding a Binomial Probabilities 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 u
22、nsecured network does not.)You randomly select 100 adults.What is the probability that exactly 35 adults believe there is no difference between secured and unsecured networks?Use technology to find the probability.(Source:University of Phoenix).21Copyright 2019,2015,2012,Pearson Education,Inc.Soluti
23、on:Finding a Binomial Probabilities Using Technology.SolutionMinitab,Excel,StatCrunch,and the TI-84 Plus each have features that allow you to find binomial probabilities.Try using these technologies.You should obtain results similar to these displays.22Copyright 2019,2015,2012,Pearson Education,Inc.
24、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 between secured and unsecured networks is about 0.012.Because 0.012 is less than 0.05,this can be considered an unusual event.
25、23Copyright 2019,2015,2012,Pearson Education,Inc.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 adults and ask them whether Google News is a major source of news for them.Find
26、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).24Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Finding Binomial Probabilities Using Formulas.25Copyright 2019,2015,2012,Pearson
27、 Education,Inc.Solution: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(0.1
28、7)2(0.83)2P(3)=4C3(0.17)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)0.26Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Finding Binomial Probabilities Using Formulas.27Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Finding Binomial Probabilities Using Formulas.28Cop
29、yright 2019,2015,2012,Pearson Education,Inc.Example:Finding a Binomial Probability Using a TableAbout 10%of workers(ages 16 years and older)in the United States commute to their jobs by carpooling.You randomly select eight workers.What is the probability that exactly four of them carpool to work?Use
30、 a table to find the probability.(Source:American Community Survey)Solution:Binomial with n=8,p=0.1,x=4.29Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Finding Binomial Probabilities Using a Table A portion of Table 2 is shownAccording to the table,the probability is 0.005.30Copyright 2019
31、,2015,2012,Pearson Education,Inc.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 considered an unusual event.31Copyright 2019
32、,2015,2012,Pearson Education,Inc.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.The
33、n graph the distribution.(Source:National Cancer Institute)Solution:n=6,p=0.62,q=0.38 Find the probability for each value of x.32Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Graphing a Binomial Distribution.Notice in the table that all the probabilities are between 0 and 1 and that the su
34、m of the probabilities is 1.33Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Graphing a Binomial DistributionHistogram:.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 0.05.34C
35、opyright 2019,2015,2012,Pearson Education,Inc.Mean,Variance,and Standard Deviation Mean:=np Variance:2=npq Standard Deviation:npq.35Copyright 2019,2015,2012,Pearson Education,Inc.Example:Mean,Variance,and Standard DeviationIn Pittsburgh,Pennsylvania,about 56%of the days in a year are cloudy.Find the
36、 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=0.44Mean:=np=300.56=16.8Variance:2=npq=300.560.44 7.4Standard Deviation:30 0.56 0.442.7npq.
37、36Copyright 2019,2015,2012,Pearson Education,Inc.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.
