《統(tǒng)計學基礎(英文版·第7版)》教學課件les7e-04-02

1、ChapterDiscrete Probability Distributions4Chapter Outline4.1 Probability Distributions4.2 Binomial Distributions4.3 More Discrete Probability Distributions.Section 4.2Binomial Distributions.Section 4.2 ObjectivesHow to determine whether a probability experiment is a binomial experimentHow to find bi

2、nomial probabilities using the binomial probability formulaHow to find binomial probabilities using technology, formulas, and a binomial probability tableHow to construct and graph a binomial distributionHow to find the mean, variance, and standard deviation of a binomial probability distribution.Bi

3、nomial ExperimentsThe experiment is repeated for a fixed number of trials, where each trial is independent of other trials.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).The probability of a success, P(S), is the s

4、ame for each trial.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 represent

5、s a 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,

6、explain why.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.Solution: Identifying and Understanding Binomial ExperimentsBinomial ExperimentEach surgery represents a trial.

7、There are eight surgeries, and each one is independent of the others.There are only two possible outcomes of interest for each surgery: a success (S) or a failure (F).The probability of a success, P(S), is 0.85 for each surgery.The random variable x counts the number of successful surgeries.Solution

8、: 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 (number of successful surgeries).Example: Identifying and Understanding Binomial ExperimentsDe

9、cide 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.A jar contains five red marbles, nine blue marbles, and six green marbles. You randomly select three marbles from the j

10、ar, without replacement. The random variable represents the number of red marbles.Solution: Identifying and Understanding Binomial 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

11、(red) for subsequent trials is no longer 5/20.The trials are not independent and the probability of a success is not the same for each trial.Binomial Probability FormulaBinomial Probability FormulaThe probability of exactly x successes in n trials isn = number of trialsp = probability of successq =

12、1 p probability of failurex = number of 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 patien

13、ts. (Source: The Orthopedic Center of St. Louis).Solution: Finding a Binomial ProbabilityMethod 1: Draw a tree diagram and use the Multiplication Rule.Solution: Finding a Binomial ProbabilityMethod 2: Use the binomial probability formula.Binomial Probability DistributionBinomial Probability Distribu

14、tionList 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)0.0160.1410.4220.422.Example: Constructing a Binomial DistributionIn a sur

15、vey, 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 num

16、ber 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.Solution: Constructing a Binomial DistributionNotice in the table that all the probabilities are between 0 and 1 and that t

17、he sum of the probabilities is 1.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;

18、 an unsecured 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).Solution: Finding a Binomial Probabilities Us

19、ing 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.Solution: Finding a Binomial Probabilities Using Technology.SolutionFrom these dis

20、plays, 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.Example: Finding Binomial Probabilities Using FormulasA survey found that 17%

21、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 the probability that (1) exactly two of them respond yes, (2) at least two of them respond yes, and (3) fewer than two of

22、them respond yes. (Source: Ipsos Public Affairs).Solution: Finding Binomial Probabilities Using Formulas.Solution: Finding Binomial Probabilities Using FormulasSolutionTo 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

23、 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(0.17)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 Formu

24、las.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 a table to find the probability.

25、 (Source: American Community Survey)Solution:Binomial with n = 8, p = 0.1, x = 4.Solution: Finding Binomial Probabilities Using a TableA portion of Table 2 is shownAccording to the table, the probability is 0.005.Solution: Finding Binomial Probabilities Using a TableYou can check the result using te

26、chnology.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.Example: Graphing a Binomial DistributionSixty-two percent of cancer survivors are ages 65 years or older. You randomly select six ca

27、ncer 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 distribution. (Source: National Cancer Institute)Solution: n = 6, p = 0.62, q = 0.38Find the probability for each value of x.Solution: Graphing a Bino

28、mial 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 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