Using-Probability-and-Pr课件

1、Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-1Business Statistics: A Decision-Making Approach6th EditionChapter 4Using Probability and Probability DistributionsBusiness Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-2Chapter GoalsAfter

2、 completing this chapter, you should be able to: nExplain three approaches to assessing probabilitiesnApply common rules of probabilitynUse Bayes Theorem for conditional probabilitiesnDistinguish between discrete and continuous probability distributionsnCompute the expected value and standard deviat

3、ion for a discrete probability distributionBusiness Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-3Important TermsnProbability the chance that an uncertain event will occur (always between 0 and 1)nExperiment a process of obtaining outcomes for uncertain eventsnElementary

4、 Event the most basic outcome possible from a simple experimentnSample Space the collection of all possible elementary outcomesBusiness Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-4Sample SpaceThe Sample Space is the collection of all possible outcomese.g. All 6 faces o

5、f a die:e.g. All 52 cards of a bridge deck:Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-5EventsnElementary event An outcome from a sample space with one characteristicnExample: A red card from a deck of cardsnEvent May involve two or more outcomes simultaneously

6、nExample: An ace that is also red from a deck of cardsBusiness Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-6Visualizing EventsnContingency TablesnTree Diagrams Red 2 24 26 Black 2 24 26Total 4 48 52 Ace Not Ace TotalFull Deck of 52 CardsRed CardBlack CardNot an AceAceAc

7、eNot an Ace Sample SpaceSample Space224224Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-7Elementary EventsnA automobile consultant records fuel type and vehicle type for a sample of vehicles2 Fuel types: Gasoline, Diesel3 Vehicle types: Truck, Car, SUV6 possible

8、elementary events: e1Gasoline, Truck e2 Gasoline, Car e3 Gasoline, SUV e4 Diesel, Truck e5 Diesel, Car e6 Diesel, SUVGasolineDieselCarTruckTruckCarSUVSUVe1e2e3e4e5e6Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-8Probability ConceptsnMutually Exclusive EventsnIf E

9、1 occurs, then E2 cannot occurnE1 and E2 have no common elementsBlack CardsRed CardsA card cannot be Black and Red at the same time.E1E2Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-9nIndependent and Dependent EventsnIndependent: Occurrence of one does not influe

10、nce the probability of occurrence of the othernDependent: Occurrence of one affects the probability of the otherProbability ConceptsBusiness Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-10nIndependent EventsE1 = heads on one flip of fair coinE2 = heads on second flip of

11、same coinResult of second flip does not depend on the result of the first flip.nDependent EventsE1 = rain forecasted on the newsE2 = take umbrella to work Probability of the second event is affected by the occurrence of the first eventIndependent vs. Dependent EventsBusiness Statistics: A Decision-M

12、aking Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-11Assigning ProbabilitynClassical Probability AssessmentnRelative Frequency of OccurrencenSubjective Probability AssessmentP(Ei) =Number of ways Ei can occurTotal number of elementary eventsRelative Freq. of Ei =Number of times Ei occursNAn opinion o

13、r judgment by a decision maker about the likelihood of an eventBusiness Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-12Rules of ProbabilityRules for Possible Values and SumIndividual ValuesSum of All Values0 P(ei) 1For any event ei1)P(ek1ii where:k = Number of elementary

14、 events in the sample spaceei = ith elementary eventBusiness Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-13Addition Rule for Elementary EventsnThe probability of an event Ei is equal to the sum of the probabilities of the elementary events forming Ei. nThat is, if:Ei =

15、e1, e2, e3then:P(Ei) = P(e1) + P(e2) + P(e3)Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-14Complement RulenThe complement of an event E is the collection of all possible elementary events not contained in event E. The complement of event E is represented by E.nC

16、omplement Rule:P(E)1)EP(EE1)EP(P(E)Or,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-15Addition Rule for Two EventsP(E1 or E2) = P(E1) + P(E2) - P(E1 and E2)E1E2P(E1 or E2) = P(E1) + P(E2) - P(E1 and E2)Dont count common elements twice! Addition Rule:E1E2+=Busines

17、s Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-16Addition Rule ExampleP(Red or Ace) = P(Red) +P(Ace) - P(Red and Ace) = 26/52 + 4/52 - 2/52 = 28/52Dont count the two red aces twice!BlackColorTypeRedTotalAce224Non-Ace242448Total262652Business Statistics: A Decision-Making

18、 Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-17Addition Rule for Mutually Exclusive EventsnIf E1 and E2 are mutually exclusive, thenP(E1 and E2) = 0SoP(E1 or E2) = P(E1) + P(E2) - P(E1 and E2)= P(E1) + P(E2)= 0E1E2 if mutually exclusiveBusiness Statistics: A Decision-Making Approach, 6e 2005 Prentic

19、e-Hall, Inc.Chap 4-18Conditional ProbabilitynConditional probability for any two events E1 , E2:)P(E)EandP(E)E|P(E221210)P(Ewhere2Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-19nWhat is the probability that a car has a CD player, given that it has AC ?i.e., we w

20、ant to find P(CD | AC)Conditional Probability ExamplenOf the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both.Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-20Conditional Probability ExampleNo CDCDTota

21、lAC.2.5.7No AC.2.1.3Total.4.61.0nOf the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both.2857.7.2P(AC)AC)andP(CDAC)|P(CD(continued)Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-21Conditional Probabili

22、ty ExampleNo CDCDTotalAC.2.5.7No AC.2.1.3Total.4.61.0nGiven AC, we only consider the top row (70% of the cars). Of these, 20% have a CD player. 20% of 70% is about 28.57%.2857.7.2P(AC)AC)andP(CDAC)|P(CD(continued)Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-22Fo

23、r Independent Events:nConditional probability for independent events E1 , E2:)P(E)E|P(E1210)P(Ewhere2)P(E)E|P(E2120)P(Ewhere1Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-23Multiplication RulesnMultiplication rule for two events E1 and E2:)E|P(E)P(E)EandP(E12121)

24、P(E)E|P(E212Note: If E1 and E2 are independent, thenand the multiplication rule simplifies to)P(E)P(E)EandP(E2121Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-24Tree Diagram ExampleDiesel P(E2) = 0.2Gasoline P(E1) = 0.8 Truck: P(E3|E1) = 0.2 Car: P(E4|E1) = 0.5 S

25、UV: P(E5|E1) = 0.3P(E1 and E3) = 0.8 x 0.2 = 0.16P(E1 and E4) = 0.8 x 0.5 = 0.40P(E1 and E5) = 0.8 x 0.3 = 0.24P(E2 and E3) = 0.2 x 0.6 = 0.12P(E2 and E4) = 0.2 x 0.1 = 0.02P(E3 and E4) = 0.2 x 0.3 = 0.06 Truck: P(E3|E2) = 0.6 Car: P(E4|E2) = 0.1 SUV: P(E5|E2) = 0.3Business Statistics: A Decision-Ma

26、king Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-25Bayes Theoremnwhere:Ei = ith event of interest of the k possible eventsB = new event that might impact P(Ei)Events E1 to Ek are mutually exclusive and collectively exhaustive)E|)P(BP(E)E|)P(BP(E)E|)P(BP(E)E|)P(BP(EB)|P(Ekk2211iiiBusiness Statistics:

27、 A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-26Bayes Theorem ExamplenA drilling company has estimated a 40% chance of striking oil for their new well. nA detailed test has been scheduled for more information. Historically, 60% of successful wells have had detailed tests, and 20% of

28、 unsuccessful wells have had detailed tests. nGiven that this well has been scheduled for a detailed test, what is the probability that the well will be successful?Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-27nLet S = successful well and U = unsuccessful welln

29、P(S) = .4 , P(U) = .6 (prior probabilities)nDefine the detailed test event as DnConditional probabilities:P(D|S) = .6 P(D|U) = .2nRevised probabilitiesBayes Theorem ExampleEventPriorProb.Conditional Prob.JointProb.RevisedProb.S (successful).4.6.4*.6 = .24.24/.36 = .67U (unsuccessful).6.2.6*.2 = .12.

30、12/.36 = .33Sum = .36(continued)Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-28nGiven the detailed test, the revised probability of a successful well has risen to .67 from the original estimate of .4Bayes Theorem ExampleEventPriorProb.Conditional Prob.JointProb.

31、RevisedProb.S (successful).4.6.4*.6 = .24.24/.36 = .67U (unsuccessful).6.2.6*.2 = .12.12/.36 = .33Sum = .36(continued)Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-29Introduction to Probability DistributionsnRandom VariablenRepresents a possible numerical value f

32、rom a random eventRandom VariablesDiscrete Random VariableContinuousRandom VariableBusiness Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-30Discrete Random VariablesnCan only assume a countable number of valuesExamples: nRoll a die twiceLet x be the number of times 4 come

33、s up (then x could be 0, 1, or 2 times)nToss a coin 5 times. Let x be the number of heads (then x = 0, 1, 2, 3, 4, or 5)Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-31Experiment: Toss 2 Coins. Let x = # heads.TTDiscrete Probability Distribution4 possible outcome

34、sTTHHHHProbability Distribution 0 1 2 x x Value Probability 0 1/4 = .25 1 2/4 = .50 2 1/4 = .25.50.25 Probability Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-32nA list of all possible xi , P(xi) pairsxi = Value of Random Variable (Outcome)P(xi) = Probability As

35、sociated with Valuenxis are mutually exclusive (no overlap)nxis are collectively exhaustive (nothing left out)n0 P(xi) 1 for each xi nS P(xi) = 1Discrete Probability DistributionBusiness Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-33Discrete Random Variable Summary Meas

36、uresn Expected Value of a discrete distribution (Weighted Average) E(x) = Sxi P(xi)nExample: Toss 2 coins, x = # of heads, compute expected value of x: E(x) = (0 x .25) + (1 x .50) + (2 x .25) = 1.0 x P(x) 0 .25 1 .50 2 .25Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.C

37、hap 4-34nStandard Deviation of a discrete distributionwhere:E(x) = Expected value of the random variable x = Values of the random variableP(x) = Probability of the random variable havingthe value of xDiscrete Random Variable Summary MeasuresP(x)E(x)x2x(continued)Business Statistics: A Decision-Makin

38、g Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-35nExample: Toss 2 coins, x = # heads, compute standard deviation (recall E(x) = 1)Discrete Random Variable Summary MeasuresP(x)E(x)x2x.707.50(.25)1)(2(.50)1)(1(.25)1)(0222x(continued)Possible number of heads = 0, 1, or 2Business Statistics: A Decision-M

39、aking Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-36Two Discrete Random VariablesnExpected value of the sum of two discrete random variables: E(x + y) = E(x) + E(y) = S x P(x) + S y P(y)(The expected value of the sum of two random variables is the sum of the two expected values)Business Statistics:

40、A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-37CovariancenCovariance between two discrete random variables: xy = S xi E(x)yj E(y)P(xiyj)where:xi = possible values of the x discrete random variable yj = possible values of the y discrete random variableP(xi ,yj) = joint probability of

41、 the values of xi and yj occurringBusiness Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-38nCovariance between two discrete random variables:xy 0 x and y tend to move in the same directionxy 5 | n = 10, p = .35) = .0949Business Statistics: A Decision-Making Approach, 6e 2

42、005 Prentice-Hall, Inc.Chap 4-59The Poisson DistributionBinomialHypergeometricPoissonProbability DistributionsDiscrete Probability DistributionsBusiness Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-60The Poisson DistributionnCharacteristics of the Poisson Distribution:nT

43、he outcomes of interest are rare relative to the possible outcomesnThe average number of outcomes of interest per time or space interval is nThe number of outcomes of interest are random, and the occurrence of one outcome does not influence the chances of another outcome of interestnThe probability

44、of that an outcome of interest occurs in a given segment is the same for all segmentsBusiness Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-61Poisson Distribution Formulawhere:t = size of the segment of interest x = number of successes in segment of interest = expected nu

45、mber of successes in a segment of unit sizee = base of the natural logarithm system (2.71828.)! xe) t()x(PtxBusiness Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-62Poisson Distribution CharacteristicsnMeannVariance and Standard Deviationt t2t where = number of successes

46、in a segment of unit sizet = the size of the segment of interestBusiness Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-63Using Poisson TablesX t00.400.500.600.700.800.90012345670.90480.09050.00450.00020.00000.00000.00000.00000.81870.16370.01640.00110.00010.0000

47、0.00000.00000.74080.22220.03330.00330.00030.00000.00000.00000.67030.26810.05360.00720.00070.00010.00000.00000.60650.30330.07580.01260.00160.00020.00000.00000.54880.32930.09880.01980.00300.00040.00000.00000.49660.34760.12170.02840.00500.00070.00010.00000.44930.35950.14380.03830.00770.00120.00020.0000

48、0.40660.36590.16470.04940.01110.00200.00030.0000Example: Find P(x = 2) if = .05 and t = 100.07582!e(0.50)! xe) t()2x(P0.502txBusiness Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-64Graph of Poisson ProbabilitiesX t =0.50012345670.60650.30330.07580.01260.00160.00020.00000

49、.0000P(x = 2) = .0758 Graphically: = .05 and t = 100Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-65Poisson Distribution ShapenThe shape of the Poisson Distribution depends on the parameters and t:t = 0.50t = 3.0Business Statistics: A Decision-Making Approach, 6e

50、 2005 Prentice-Hall, Inc.Chap 4-66The Hypergeometric DistributionBinomialPoissonProbability DistributionsDiscrete Probability DistributionsHypergeometricBusiness Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-67The Hypergeometric Distributionn“n” trials in a sample taken f

51、rom a finite population of size NnSample taken without replacementnTrials are dependentnConcerned with finding the probability of “x” successes in the sample where there are “X” successes in the populationBusiness Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-68Hypergeome

52、tric Distribution FormulaNnXxXNxnCCC)x(P.WhereN = Population sizeX = number of successes in the populationn = sample sizex = number of successes in the sample n x = number of failures in the sample(Two possible outcomes per trial)Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall

53、, Inc.Chap 4-69Hypergeometric Distribution Formula0.3120(6)(6)CCCCCC2)P(x1034261NnXxXNxn Example: 3 Light bulbs were selected from 10. Of the 10 there were 4 defective. What is the probability that 2 of the 3 selected are defective? N = 10n = 3 X = 4 x = 2Business Statistics: A Decision-Making Appro

54、ach, 6e 2005 Prentice-Hall, Inc.Chap 4-70Hypergeometric Distribution in PHStatnSelect:PHStat / Probability & Prob. Distributions / Hypergeometric Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-71Hypergeometric Distribution in PHStatnComplete dialog box entries

55、 and get output N = 10 n = 3X = 4 x = 2 P(x = 2) = 0.3(continued)Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-72The Normal DistributionContinuous Probability DistributionsProbability DistributionsNormalUniformExponentialBusiness Statistics: A Decision-Making App

56、roach, 6e 2005 Prentice-Hall, Inc.Chap 4-73The Normal Distributionn Bell Shapedn Symmetrical n Mean, Median and Mode are EqualLocation is determined by the mean, Spread is determined by the standard deviation, The random variable has an infinite theoretical range: + to Mean = Median = Modexf(x)Busin

57、ess Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-74By varying the parameters and , we obtain different normal distributionsMany Normal DistributionsBusiness Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-75The Normal Distribution Shapexf(x)Chan

58、ging shifts the distribution left or right.Changing increases or decreases the spread.Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-76Finding Normal Probabilities Probability is the area under thecurve!abxf(x)P axb() Probability is measured by the area under the

59、curveBusiness Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-77f(x)xProbability as Area Under the Curve0.50.5The total area under the curve is 1.0, and the curve is symmetric, so half is above the mean, half is below1.0)xP(0.5)xP(0.5)xP(Business Statistics: A Decision-Maki

60、ng Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-78Empirical Rules 1 encloses about 68% of xs f(x)x1111What can we say about the distribution of values around the mean? There are some general rules:68.26%Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 4-79The Empirical Rul